Secondary Market Liquidity and the Optimal Capital Structure

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Secondary Market Liquidity and the Optimal Capital Structure David M. Arseneau, David E. Rappoport, and Alexandros Vardoulakis 2015-031 Please cite this paper as: David M. Arseneau, David E. Rappoport, and Alexandros Vardoulakis (2015). “Secondary Market Liquidity and the Optimal Capital Structure,” Finance and Economics Discussion Series 2015-031. Washington: Board of Governors of the Federal Reserve System, http://dx.doi.org/10.17016/FEDS.2015.031. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Secondary Market Liquidity and the Optimal Capital Structure* David M. Arseneau David E. Rappoport Alexandros P. Vardoulakis FederalReserveBoard FederalReserveBoard FederalReserveBoard May12,2015 Abstract We present a model where endogenous liquidity generates a feedback loop between secondary market liquidity and firms’ financing decisions in primary markets. The model features two key frictions: a costly state verification problem in primary markets, and search frictions in over-the-counter secondary markets. Our concept of liquiditydependsendogenouslyonilliquidassetsputupforsalerelativetotheresources available for buying those assets in the secondary market. Liquidity determines the liquiditypremium,whichaffectsissuanceintheprimarymarket,andthiseffectfeeds backintosecondarymarketliquiditybychangingthecompositionofinvestors’portfolios. Weshowthattheprivatelyoptimalallocationsareinefficientbecauseinvestors and firms fail to internalize how their behavior affects secondary market liquidity. Theseinefficienciesareestablishedanalyticallythroughasetofwedgeexpressionsfor keyefficiencymargins. Ouranalysisprovidesarationalefortheeffectofquantitative easingonsecondaryandprimarycapitalmarketsandtherealeconomy. Keywords: Marketliquidity,secondarymarkets,capitalstructure,quantitativeeasing. JELclassification: E44,G18,G30. * WearegratefultoFrancescaCarapella, GiovanniFavara, NobuKiyotaki, CeciliaParlatore, LasseH. Pedersen,SkanderVandenHeuvelandseminarparticipantsatFederalReserveBoardandCowlesGeneral Equilibrium Conference for comments. All errors herein are ours. The views expressed in this paper are thoseoftheauthorsanddonotnecessarilyrepresentthoseofFederalReserveBoardofGovernorsoranyone intheFederalReserveSystem. Emails: david.m.arseneau@frb.gov,david.e.rappoport@frb.gov,alexandros.vardoulakis@frb.gov. 1

1 Introduction Secondarymarketliquidityisanimportantconsiderationforinvestorsbuyinglong-term assets. Atthesametime,theissuanceoflong-termdebtinprimarymarketsaffectsmarket liquidity by altering the maturity composition of investors’ portfolios. The interaction between primary debt markets and secondary market liquidity is important for understanding the real effects of financial market imperfections. For example, how does debt issuance in primary markets affect liquidity in secondary markets? How does investors’ demand to be compensated for bearing liquidity risk affect the firm’s incentive to issue debt in the primary market? Does the interaction between these two channels lead to an efficient capital structure of the firm? How does quantitative easing affect the real economythroughinterventionineithertheprimaryorsecondarymarket? This paper presents a model to formalize the interaction between primary and secondary capital markets in order to shed light on these questions. In particular, we are interested in imperfect secondary trading that gives rise to liquidity risk, as investors’ liquidityneedscannotbemetbysellingassetsfrictionlesslyinsecondarymarkets. Wemakethreemaincontributions. First,weuncoveranovelfeedbackloop,illustrated in Figure 1, between secondary market liquidity and the firm’s financing decision in primary capital markets. This feedback loop allows for liquidity risk associated with trade in the secondary market to influence firms’ financing decisions through funding costs.1 This direct channel has received considerable attention in the literature as it is closely related to the idea of transaction or information costs impeding trading, as well to the lending channel of monetary policy. Our framework differs, however, in that we capture an additional channel whereby the firm’s financing decisions in the primary market feed back into the determination of liquidity in the secondary market. This happens both directly through the supply of long-term assets and indirectly by altering thecompositionofinvestorportfolios. Thislinkbetweenprimaryissuanceandsecondary marketliquidityhasnotbeenstudiedintheliterature,butitiskeytounderstandinghow theliabilitystructureoffirmsmattersfortheoptimalintermediationofliquidityriskand therealeconomy. Weprovetheexistenceanduniquenessofanequilibriumfeaturingthis feedbackloopandcharacterizeitinclosedform. Our second main contribution is to show that this feedback loop distorts capital markets. Theinteractionbetweentheprimaryandsecondarymarketsleadstotwodistortions: 1In a seminal paper, Holmström and Tirole (1998) study a similar question to ours, but focus on the liquidityneedsoffirmstocoveroperationalcostsbeforetheirinvestmentmatures. Incontrast,wefocuson theliquiditydemandoflenders. Tothisextent,wemodelthedemandforliquidityasintheseminalpaper ofDiamondandDybvig(1983),butbringre-tradingoflong-termassets,aggregateliquidityandthecapital structuretothecenterofouranalysis. 2

Lendersimposeliquiditypremia PrimaryMarket SecondaryMarket Borrowingaffectsliquidity Figure1: Feedbackloopbetweenprimaryandsecondarymarketforcorporatedebt oneinthecapitalstructureofthefirmandanotherintheallocationofinvestorportfolios. Thesedistortionsarisefromthefactthatneitherfirmsnorinvestorsinternalizehowtheir behavior affects liquidity in the secondary market. In equilibrium, market liquidity can besuboptimallylow(high)implyingthefirmisover-leveraged(under-leveraged),hence there is an under-supply (over-supply) of liquid assets for investors trading on the secondary market. A social planner would like to implement the optimal level of liquidity in the secondary market by altering the financing decisions of firms and the portfolio allocations of investors. Such an outcome leads to higher firm profits while investors are no worse off. We derive a set of analytic wedge expressions that highlight two distorted marginsandshowhowanappropriatelydesignedtaxsystemcandecentralizetheefficient equilibrium. Our third contribution is to provide a theoretical characterization for the effects of quantitative easing (QE) policies, like the ones observed following the Great Recession. Throughthelensofourmodel,policiesthataffectthecompositionofinvestors’portfolios, suchasquantitativeeasing,affecttheeconomybycompressingliquiditypremia,thereby influencing savings and investment decisions in the real economy (see Stein, 2014, for a general discussion). Our analysis also highlights the benefits and limitations of such interventions. On the one hand, QE can improve the intermediation capacity of the economy by expanding its productive frontier. On the other hand, these policies may be limited by their redistributive effects, the disadvantage of central banks in monitoring borrowers,andtheprospectforfinanciallosses. Themodelhasthreeperiods,anditispopulatedbyfirmsthatneedexternalfinancing to invest in long-term projects and investors who want to transfer funds over time to consume in all periods. In the initial period, ex ante identical investors supply funds to firmsinprimarycapitalmarkets,whilefirmsissueclaimsagainsttheirlong-termrevenues that materialize only in the final period. The contracting problem between the firm and investorsintheprimarydebtmarketissubjecttoagencyfrictions,whichwemodelusing the costly state verification (CSV) framework (Townsend, 1979; Gale and Hellwig, 1985; 3

Bernanke and Gertler, 1989). The choice of the CSV framework is guided by the fact that it offers a convenient and well understood rationale for the firm’s use of debt financing, which is central to our model. In addition, the CSV framework allows us to jointly study theeffectofliquiditypremiaonthecomposition(leverage)andtheriskinessofthecapital structure of the firm. That said, the specific nature of the agency frictions in the primary marketisnotdetrimentalforthegeneralityofourresults.2 After the financial contract between the firm and investors is written and investment decisionsaremade,asubsetofinvestorsreceiveidiosyncratic(liquidity)shocksthatmake themwanttoconsumebeforethefirm’sinvestmentsmatureandproceedsaredistributed. These shocks are private information and, thus, contingent contracts among patient and impatient investors cannot be written ex ante. Alternatively, investors can self-insure by investingpartoftheirendowmentinastoragetechnologyorbyholdingcorporatebonds and re-trading them in a secondary market once the type has been revealed. Corporate bonds thus not only are a claim on real revenues, but also have a role in facilitating exchange(seealsoRocheteauandWright,2013). In absence of frictions, the ability to trade long-term bonds in the secondary market would perfectly satisfy impatient investors’ demand for liquidity. Indeed, in this special case we show that our model collapses to the benchmark CSV model of Bernanke and Gertler(1989)whereliquidityconcernsplaynorole. Inpractice,however,tradingfrictions may impinge on the ability of impatient investors to sell long-term assets. For corporate bonds, which are traded in over-the-counter (OTC) markets, empirical evidence by Edwards et al. (2007) and Bao et al. (2011) suggests that search frictions are an important driverofliquiditypremia.3 2The reason is that we are able to disentangle the channel through which market liquidity affects liquidity premia in long-term assets from the choice of the optimal contract/capital structure of the firm. Hence, itdoesnotmatterhowweintroducethefinancingfrictions. Forexample, asituationwherefirms facecollateralconstraintsasinHolmströmandTirole(1997) orKiyotakiandMoore(1997)wouldyieldthe samequalitativeresults. Thatsaid, thereisafundamentaldifferencebetweenmodelsfeaturingcollateral constraints and our framework with respect to the concept of liquidity. In the language of Brunnermeier andPedersen(2009),theformeremphasizesfundingliquidity(howmuchfirmscanraisebypledgingassets ascollateral),whileourtheoryhighlightstheimportanceofmarketliquidity(theeasewithwhichilliquid assetscanbesold). 3Bond financing has become one of the most important sources of external financing for U.S. corporations. Figure 3 shows that bond financing is the dominant source of credit liabilities for non-financial corporate firms (Financial Accounts of the United States data). This paper focuses on bond financing abstracting from the fact that firms enter into bank loans or other types of borrowing at the same time (see deFioreandUhlig, 2011, AokiandNikolov, 2014, formodelswherebankandbondfinancingcoexist). In principle,bankintermediationwouldbeoptimaltoinsureagainstidiosyncraticliquidityriskinthespiritof DiamondandDybvig(1983)whenbankrunsarenotverylikely(seeCooperandRoss,1998,andGoldstein andPauzner,2005)orbankcreditisnotsufficientlymoreexpensivethanbondfinancingasindeFioreand Uhlig(2011). However,Jacklin(1987)showsthattheefficiencygainsofbankintermediationforinvestors vanish when secondary capital markets are available and function frictionlessly. This should continue to be true when the associated frictions in secondary markets are not too severe, while bank intermediation 4

We follow Duffie et al. (2005), Lagos and Rocheteau (2007, 2009), and others by introducing illiquidity in the secondary market through search frictions between buyers and sellers that engage in OTC trade. In our framework, impatient investors submit sell orders that are matched with buy orders submitted by patient investors through a matchingfunction. Theefficiencyofthematchingtechnologyinfluencesthelikelihoodof trading opportunities for both buyers and sellers in a symmetric fashion. Additionally, our framework allows trade probabilities to be endogenously determined by market liquidity, defined as the number of buy orders relative to sell orders. This notion of market liquidity will have an asymmetric effect on trading opportunities for buyers relative to sellers. Hence, our approach is distinct from most of the existing literature studying search frictions in OTC markets, which treats matching probabilities as exogenous.4 Moreover, mostofthisliteraturefocusesontheimplicationsofsearchfrictionsandilliquidityspecifically on asset prices. Price effects are important in our framework as well, but our focus isbroaderinthesensethatweareinterestedinhowprimarymarketsforcorporateassets interactwithsecondarymarketliquidity. Beforeturningtothedetailsofthemodel,weshouldnotethatwehaveabstractedaway fromissuesrelatedtoadverseselectionarisingfromasymmetricallyinformedagentsparticipating in the secondary market. In a seminal paper, Gorton and Pennacchi (1990) show how the information sensitivity of financial contracts affects their liquidity in secondary markets and study the capital structure of the firm and efficient intermediation.5 woulddominatewhenmarketsaremoreimperfect(Diamond,1997). 4Duffie et al. (2005) assume the holdings of agents participating in the OTC markets do not play an importantroleinequilibriumoutcomes. LagosandRocheteau(2009)utilizethefactthatagentscanmitigate tradingfrictionbyadjustingtheirassetpositiontoreducetheirtradingneeds. Thus, theycanstudyhow liquiditypremiaaffecttheportfolioholdingsofagents,butnotthereverselinkagefromportfoliostomarket liquidity. HeandMilbradt(2014)presentamodelwithsearchfrictionsinOTCmarketsforcorporatebonds and show how default and liquidity premia, as well as the decision to default, are affected by market liquidity. However,theytakethecapitalstructureandinvestmentofthefirmasgiven,whichinourmodel isendogenousandattheheartofouranalysis. BrucheandSegura(2014)endogenizetheratioofbuyersto sellersbyallowingfreeentryofpatientinvestors,whobringnewresourcesintheeconomy,andstudyhow the entry decision interacts with the efficient choice of debt maturity given fixed firm size. Our concept of liquidity differs as it is endogenous even without free entry. Geromichalos and Herrenbrueck (2015) examinehowOTCmarketsandliquidityaffectassetpricesinamoneysearchmodelofLagosandWright (2005). 5There is an important literature following this tradition, such as Dang et al. (2011) and Gorton and Ordoñez (2014). Guerrieri and Shimer (2014) examine how adverse selection about the quality of assets affects their liquidity premia. They differ from the search microfoundations of illiquidity because the difficulty of finding a buyer depends primarily on the extent of private information rather than the availability of trading opportunities. Like us, but for different reasons, they suggest that unconventional policyinterventions,suchasassetpurchase,canenhancetheliquidityofassetsnotincludedinthepurchase programs. Nevertheless,theydonotstudyhowilliquidityandpolicyinterventionsaffecttheequilibrium supply of assets, i.e. they abstract from corporate finance issues. Malherbe (2014), who builds on an adverseselectionmodelofliquiditybyEisfeldt(2004),showsthat,incontrast,excesscash-holdingsimpose 5

Although similar in spirit, our approach differs with respect to the frictions resulting in illiquid liabilities of the firm. Gorton and Pennacchi (1990) show that uninformed investors respond by demanding informationally insensitive assets, notably riskless debt. Hence,theirapproachisimportantforunderstandinghowinvestors’decisionstoparticipateinthesemarkets(theextensivemarginofinvestors’portfoliochoice)affectsthefirm’s capital structure. In contrast, our approach of introducing search frictions to limit trade in secondary markets allows us to examine how—given full participation in both asset markets—the intensive margin of investors’ portfolio choice affects the firm’s financing decisionandhowthefirm’sfinancingdecision,inturn,affectsinvestors’portfolios. We have also abstracted away from aggregate liquidity risk. When investors face aggregateliquidityriskwhichcannotbehedgedduetomarketincompleteness,liquidity provision in the form of aggregate savings/reserves may be suboptimally low (Bhattacharya and Gale, 1987; Allen and Gale, 2004).6 In our paper, inefficient liquidity stems from trading frictions rather than aggregate shocks, which yields important implications fortheliquiditypremiaofcorporatebondsduringperiodsthataggregateliquidityshocks areexpectedtooccurratherinfrequently. Consequently,ourmechanismcouldpotentially explainthefluctuationsinliquidityanddefaultriskpremia,aswellasfirms’leverageeven when aggregate liquidity shortages are unlikely or excluded due to the presence of unconventionalpolicies,suchasquantitativeeasing. Moreover,ourmechanismcanalsorationalizesituationswheretheremaybeanoverprovision of liquidity in the market economy. The reason is that our trading frictions do not only matter for the sellers of assets in the secondary market, who benefit from high liquidity,butalsoforthebuyers,whoaremorelikelytoextractrentswhenliquidityislow. HartandZingales(2015)showthatthelackofadoublecoincidenceofwantscanresultin apenuniaryexternalityoperatingthroughtherelativepriceoftradedgoodsandservices, andrenderprivateliquidityholdingsinefficientlyhigh. Inourmodel,inefficientliquidity in general does not accrue from a relative price externality or a fire-sale, but rather from therelativeeasinessforbuyersandsellerstotrade. The rest of the paper proceeds as follows. Section 2 presents the model and derives the equilibrium conditions. Section 3 shows how secondary market liquidity interacts with the optimal financing decisions of the firm. Section 4 present the social planner’s problem, and identifies the externalities inherent in the private economy as well as the a negative externality on others because they reduce the quality of assets put for sale in the secondary market. SeealsoKurlat(2013)andBigio(2015)fortheinteractionofbusinesscycledynamicsandilliquidity inducedbyadverseselectioninassetmarkets. 6Liquidityunder-provisionmayalsostemsfromhiddentradesundoingtheefficientsharingofliquidity riskacrossimpatientandpatientagentsasinFarhietal.(2009)orfire-salesexternalities(Lorenzoni,2008; Korinek,2011;Acharya,ShinandYorulmazer,2011),whichweabstractfrominourpaper. 6

optimal policy mix. Section 5 analyses the effect of quantitative easing on secondary market liquidity and financing decisions. Finally, section 6 concludes. All proofs are relegatedtotheAppendix. 2 Model 2.1 Physical Environment Therearethreetimeperiodst = 0,1,2,asingleconsumptiongood,andtwotypeofagents: entrepreneursandinvestors. Entrepreneurshavelong-terminvestmentprojectsandmay fund these projects with internal funds or with loans from investors. Ex ante identical investors lend funds to entrepreneurs, but once that lending has taken place and while production is underway, investors are subject to a preference (liquidity) shock which revealswhethertheyareimpatient,andhenceprefertoconsumeearlierratherthanlater, orpatient. Thesetwotypesofinvestorstradetheirassetsinsecondaryassetmarketswith searchfrictions(seeFigure 2). Primarydebtmarket Firm undertakes a long-term risky investment project Firm Firm SecondaryOTCmarket Patient 1 δ Investor − Illiquid Liquid Investor Investor asset asset Impatient δ Uncertainty, ω, is realized; Investor risky project pays out Some investors are hit with a liquidity shock t = 0 t = 1 t = 2 Figure2: Timeline. 7

Thereisamassoneofexanteidenticalentrepreneurs,whoareendowedwith n units 0 of capital at t = 0. Entrepreneurs invest to maximize the return on their equity, i.e., to maximize profits per unit of endowment. The technology is linear and delivers Rkω at t = 2, per unit invested at t = 0. The random variable ω is an idiosyncratic productivity shock that hits after the project starts, and is distributed according to the cumulative distribution function F, with unit mean. It is privately observed by the entrepreneur, but investors can learn about it when they seize entrepreneurs’ assets and pay a monitoring costs μ as a fraction of assets. The (expected) gross return Rk is assumed to be known at t = 0, as there is no aggregate uncertainty in the model. In order to produce, the firm mustfinanceinvestment,denoted k ,eitherthroughitsownfundsorbyissuingfinancial 0 contracts to investors. So profits equal total revenue in period 2, Rkωk , minus payment 0 obligations from financial contracts. Entrepreneurs represent the corporate sector in our model,sowewilltalkaboutentrepreneurs’projectsandfirmsinterchangeably. There is a mass one of ex ante identical investors, who are endowed with e units of 0 capital at t = 0. Investors have unknown preferences at t = 0, and learn their preferences at t = 1. At t = 1 investors realize if they are patient or impatient consumers, a fraction 1 δ will turn out to be patient and a fraction δ impatient. Patient consumers have − preferences only for consumption in t = 2, uP(c ,c ) = c , whereas impatient consumers 1 2 2 havepreferencesforbothconsumptionin t = 1and2,butdiscountperiod2consumption atrate β,uI(c ,c ) = c +βc . 1 2 1 2 Investors in both period 0 and 1 have access to a storage technology with yield r > 0, i.e., every unit stored yields 1 + r units of consumption in the next period. The amount stored in period t is denoted s . In addition, at t = 0, they can invest in financial contracts t issued by entrepreneurs in primary markets; and, at t = 1, they can buy and sell assets in secondary markets with search frictions (see Figure 2). When engaging in trade in the secondary market patient investors realize a return Δ. Both the primary and secondary marketsaredescribedindetailbelow.7 Inwhatfollowswemakethefollowingassumptions. Assumption 1 (Relative Returns) The long-term return of the productive technology is larger than the cumulative two-period storage return and the return on storage plus the return on secondary markets, i.e., (1+r)2 < Rk and (1+r)Δ < Rk. In addition, monitoring costs are such thatRk(1 μ) < (1+r)2. − Assumption 2 (Productivity Distribution) Let h(ω) = dF(ω)/(1 F(ω)) denote the hazard − rateoftheproductivitydistribution. Itisassumedthat ωh(ω) isincreasing. 7Note that since r > 0 and since investors preferences have been assumed time separable and risk neutral, there was no loss of generality in abstracting away from consumption at t = 0 for investors, and consumptionatt=1forpatientinvestors. 8

Assumption 3 (Impatience) The rate of preference of impatient investors is such that β ≤ 1/(1+r). Assumption4(InvestorsDeepPockets) Itisassumedthatinvestors’(total)endowmente is 0 significantlyhigherthanentrepreneurs’(total)endowmentn ,i.e.,e >> n . 0 0 0 Assumption 1 is necessary for there to be a role for the entrepreneurial sector, Rk > (1 + r)2, and, Rk > (1 + r)Δ, when the prospective return on secondary market is taken into account. Furthermore, this assumption rules out equilibria where entrepreneurs are always monitored, (1 + r)2 > Rk(1 μ). Assumption 2 ensures that there is no credit − rationing in equilibrium, and together with Assumption 1 will ensure the existence and uniquenessofequilibrium,aswediscussbelow. Assumption3makesimpatientinvestors havea(weak)preferenceforcurrentversusfutureconsumptionwhentheinterestrateis r. Finally,Assumption4ensuresthatinvestorscanmeetthecreditdemandofentrepreneurs. 2.2 The Financial Contract Entrepreneurs finance their investments using either internal funds, n , or by selling 0 long-term financial contracts to investors in the primary corporate debt market. These contracts specifie an amount, b , borrowed from investors at t = 0 and a promised gross 0 interest rate, Z, made upon completion of the project at t = 2. If entrepreneurs cannot make the promised interest payments, investors can take all firm’s proceeds paying a monitoringcost,equaltoafraction μ ofthevalueofassets.8 The t = 0budgetconstraintfortheentrepreneurisgivenby k n +b . (1) 0 0 0 ≤ For what follows it will be useful to define the entrepreneur’s leverage, l , as the ratio of 0 assetsto(internal)equity k /n . 0 0 Theentrepreneurisprotectedbylimitedliability,soitsprofitsarealwaysnon-negative. Thus,theentrepreneur’sexpectedprofitinperiod t = 2isgivenby E max 0,Rkωk Zb . 0 0 0 − n o Limited liability implies that the entrepreneur will default on the contract if the realization of ω is sufficiently low such that the payoff of the long-term project falls below 8We consider deterministic monitoring rather than stochastic monitoring, which results in debt being theoptimalcontract. KrasaandVillamil(2000)derivetheconditionsunderwhichdeterministicmonitoring occurs in equilibrium in costly enforcement models. In addition, our model features perfect, but costly, ex-post enforcemnt. See Krasa et al. (2008) for a more elaborate enforcement process and its implications forfirms’finance. 9

the promised payout; that is, when Rkωk < Zb . This condition defines a threshold 0 0 productivitylevel, ωˉ,suchthattheentrepreneurdefaultswhen Z l 1 0 ω < ωˉ = − . (2) Rk l 0 Theproductivitythresholdmeasuresthecreditriskofthefinancialcontract;andisincreasinginthespreadbetweenthepromisedreturnandtheexpectedreturnontheentrepreneur investment,andincreasinginfirm’sleverage. ωˉ Fornotationalconvenience,wedefineG(ωˉ) ωdF(ω)andΓ(ωˉ) ωˉ(1 F(ωˉ))+G(ωˉ). ≡ 0 ≡ − The function G(ωˉ) equals the truncated expectation of entrepreneurs’ productivity given R default. The function Γ(ωˉ) equals the expected value of a random variable equal to ω if thereisdefault(ω < ωˉ)andequalto ωˉ whenthereisnot(ω ωˉ). Itfollowsthat Rkk Γ(ωˉ) 0 ≥ correspondstotheexpectedtransfersfromentrepreneurstoinvestors. Then, firms’ objective, expected profits per unit of endowment, or return on equity, canbeexpressedusingthepreviousnotationas9 1 E max 0,Rkωk Zb = [1 Γ(ωˉ)]Rkl . (3) 0 0 0 0 n − − 0 n o Similarly,the total expectedpayoff ofbondcontractscanbeexpressedas ωˉ ∞ Zb dF(ω)+(1 μ) Rkωk dF(ω) = k Rk Γ(ωˉ) μG(ωˉ) . 0 0 0 − − Zωˉ Z0 (cid:2) (cid:3) Therefore,theexpectedgrossreturnofholdinga single bondtomaturity Rb isgivenby l Rb = 0 Rk Γ(ωˉ) μG(ωˉ) , (4) l 1 − 0 − (cid:2) (cid:3) whichisafunctionofonlyleverageandtheproductivitythreshold. Clearly Rb is decreasing in l as leverage dilutes lenders claim on the firm’s assets. 0 Moreover, in equilibrium it will be increasing in risk, ωˉ, as detailed below. Finally, note that the expected return is known in period 0 and 1, since there is no aggregate uncertainty or new information arriving after investors and the firm have agreed on the termsoflending. Thismeansthatidiosyncraticliquidityshocksinperiod1donotaffectRb andinvestorswouldtradebondsinasecondarymarketpromisingthisexpectedpayout. 9Theobjectiveofthefirminequation(3)iswrittenintermsofreturntoequityratherthantotalprofits. However,bothformulationswouldyieldthesameequilibriumresultsas n ispositiveandgiven. 0 10

2.3 The Secondary OTC Market The ex post heterogeneity introduced by the preference shock generates potential gains fromtradingcorporatedebtinasecondarymarket. Impatientinvestorswanttoexchange long-term, imperfectly liquid, bonds for consumption, as they would rather consume at theendofperiod1thanholdthebondtomaturityuntilperiod2(Assumption 3). Patient investors are willing to exchange lower yielding storage for corporate debt with higher expectedreturns. Inorderforsuchatradetotakeplace,buyandsell ordersmustbepairedupaccording toamatchingtechnologywhichalignsthem. Impatientinvestorssubmitsaleorders,one for each bond they are ready to sell at a given price q . Patient investors submit buy 1 orders, one for each package of q units of storage they are ready to exchange for a bond. 1 We model the OTC market such that matching is by order, as opposed to by investor.10 Suppose, in aggregate, there are A sell (or ask) orders and B buy orders. The matching functionisassumedtobeconstantreturnstoscaleandisgivenby m(A,B) = νAαB1 α , (5) − with 0 < ν a scaling constant and 0 < α < 1 the elasticity of the matching function with respect to sell orders. The number of matches is limited by the minimum of the number ofbuyandsellorders,so m(A,B) min A,B . ≤ { } Wedefineaconceptofmarketliquiditythroughtheratioofbuyorderstosellorders,or θ = B/A. Thisnotionofliquidity—definedbyaconceptofthicknessintheOTCmarket— hasdifferentimplicationsfortradersonopposingsidesofthemarket. Forexample,when θ is large, a bond in the secondary market is relatively liquid, that is, it is relatively easy for sellers to trade. But, at the same time, it is relatively hard for buyers to trade. Note that our notion of liquidity is related to, but distinct from, the easiness to trade for all marketparticipants,whichiscapturedinourframeworkbytheefficiencyofthematching technology ν. Increasing (decreasing) ν makes it easier (harder) for participants on both sidesofthemarkettotradeinasymmetricfashion. Usingthematchingfunction,the probabilitythatasellorderisexecuted isexpressedas m(A,B) f(A,B) = or f(θ) = m(1,θ) , (6) A andthe probabilitythatabuyorderisexecuted isexpressedas m(A,B) p(A,B) = or p(θ) = m(θ 1,1) . (7) − B 10Thiscanbethoughofasmoneychasingbonds,insteadofinvestorschasinginvestors. 11

The fact that matches are bounded by the minimum number of orders, i.e., m(A,B) ≤ min A,B ,definestwoliquiditythresholdθandθ. Whenliquidityissmallerthanθ = ν1/α { } thenallbuyordersareexecuted,i.e.,m(A,B) = B. Inthiscasebuyerstradewithprobability p(θ) = 1, whereas sellers trade with probability f(θ) = θ. Alternatively, when liquidity is higher than θ = ν 1/(1 α) then all sell orders are executed, i.e., m(A,B) = A; and thus − − the trade probabilities f(θ) = 1 and p(θ) = θ 1. When liquidity is in [θ,θ] then matches − are given by the matching function (5) and the trade probabilities by equations (6) and (7). Unless otherwise stated, we restrict attention to the case ν < 1, which guaranties that θ < θ. Onceabuyorderandasellorderarematched,thetermsoftradearedeterminedviaa simplesurplussharingruleknownbyallagents. Fromtheseller’sperspective,atrading match yields additional liquid wealth from unloading the incremental bond sold at price q . Ifthesellerwalksawayfromthematchsheholdsthebond,whichmaturesinthefinal 1 period, delivering an expected payout of Rb in t = 2, which is discounted at rate β. The value of a trading match to a buyer is the present value of the (expected) return on the bond,netofthepricethatneedstobepaidforeachbondinthesecondarymarket. Then, the surplus that accrues to an impatient investor, SI(q ), and the surplus that accrues to a 1 patientinvestor, sP(q ),respectively,aregivenby 1 Rb SI(q ) = q βRb and SP(q ) = q . 1 1 1 1 − 1+r − Thepriceofthedebtcontractonthesecondarymarketisdeterminedbyasharingrule thatmaximizestheNashproductoftherespectivesurpluses, ψ 1 ψ max SI(q ) SP(q ) − , 1 1 q1 (cid:16) (cid:17) (cid:16) (cid:17) where ψ [0,1] is a parameter that determines the split of the surplus between patient ∈ andimpatientinvestors.11 The solution of the surplus splitting problem yields the following bond price in the secondarymarket ψ q = Rb +(1 ψ)β . (8) 1 1+r − ! Note that ψ = 1 drives the price of the bond to the “bid” price, or the price that extracts fullrentfromthebuyer, q = Rb . Bythesametoken, ψ = 0drivesthepriceofthebondto 1 1+r the“ask”price,orthepricethatextractsfullrentfromtheseller, q = βRb. Fromequation 1 11Our sharing rule is very close to Nash bargaining over the surplus. Under Nash bargaining the parameterψcanbeinterpretedasthebargainingpowerofsellers. 12

(8) it follows that the return that patient investors make in the secondary market, per executedbuyorder,dependsonlyonexogenousparametersandisgivenby 1 Rb ψ − Δ = = +(1 ψ)β . q 1+r − 1 ! 2.4 Investors As described above, investors are ex ante identical and are endowed with e units of 0 capital. At t = 0 they can allocate their wealth across two assets: the storage technology anddebtcontracts. Thus,theirbudgetconstraintisgivenby12 s +b = e , (9) 0 0 0 where s ,b 0, i.e. borrowing at the storage rate or short-selling corporate debt are not 0 0 ≥ allowed. The storage technology, denoted s , pays a fixed rate of return 1+r at t = 1 in units of 0 consumption. Theproceedsofthisinvestment,ifnotconsumed,canbereinvestedtoearn an additional return of 1+r between period 1 and 2, again paid in units of consumption. In this sense, storage is a liquid investment, as at any point in time it can be costlessly transformedintoconsumption. Alternatively,thecorporatebondhasanexpectedpayoff of Rb, but only at the beginning of t = 2. Moreover, for an investor to turn her bond into consumption at t = 1, she will have to post an order in a secondary market characterized bysearchfrictions. Sothebondisilliquid,asitdoesnotallowinvestorstotransformtheir investmentcostlesslyintoconsumptioninperiod1. The relative illiquidity of corporate debt comes into play because at the beginning of t = 1,afractionδofinvestorsreceiveapreferenceshockthatmakesthemdiscountfuture consumption at rate β. Moreover, Assumption 3 implies that impatient investors prefer to consume in period 1 relative to period 2. In contrast, the remaining fraction 1 δ are − patientinvestors,whoonlyenjoyconsumptionin t = 2. Thus, impatient investors find themselves holding corporate debt contracts which cannot easily be transformed into period t = 1 consumption. Ideally, they would like to sell this asset to patient investors who are willing to give up units of liquid storage in exchange for the higher yielding corporate debt. This trading activity takes place in an OTC secondary market. As described above, impatient investors looking to unload corporatedebtcontractswillonlygettheirordersexecutedwithendogenousprobability f(θ). Similarly, patient investors looking to purchase corporate debt will only get their 12Sincethemassofbothentrepreneursandinvestorsequalsone,andwefocusonthesymmetricequilibrium,weabusenotationanddenotetheindividualsupplyanddemandofdebtby b . 0 13

orders executed with endogenous probability p(θ). If a buy and a sell order are lucky enoughtobematchedintheOTCmarketabilateraltradetakesplaceandunitsofbonds areexchangedforunitsofstorageattheagreeduponprice q . 1 To describe the portfolio choice problem of investors, it is useful to first consider the optimal behavior of impatient and patient investors in t = 1 when they arrive to that periodwithagenericportfolioofstorageandbonds(s ,b ). 0 0 2.4.1 ImpatientInvestors By Assumption 3 at t = 1 impatient investors want to consume in the current period. They can consume the payout from investing in storage, s (1 + r), plus the additional 0 proceeds from placing b sell orders in the OTC market. These orders are executed 0 with probability f(θ) and each executed order yields q units of consumption. Thus, the 1 expectedconsumptionofimpatientinvestorsinperiod1isgivenby cI = s (1+r)+ f(θ)q b . (10) 1 0 1 0 On the other hand, with probability 1 f(θ) orders are not matched and impatient − investorsareforcedtocarrydebtcontractsintoperiod2. Therefore,expectedconsumption inthefinalperiodisgivenby cI = (1 f(θ))Rbb , (11) 2 − 0 andtheutilityderivedfrom cI isdiscountedby β. 2 2.4.2 PatientInvestors Patient investors only value consumption in the final period and will be willing to place buy orders in the OTC market if there is a surplus to be made, i.e., if q Rb/(1+r). The 1 ≤ pricedeterminationintheOTCmarketguaranteesthatthisisalwaysthecase(1 +r Δ), ≤ thus patient investor would ideally like to exchange all of the lower yielding units of storage for corporate debt with a higher expected returns. But their buy orders will be executedonlywithprobability p(θ). Therefore, expected storage holdings at the end of t = 1, sP, are equal to a fraction 1 1 p(θ)oftheavailableliquidfunds s (1+r),i.e., 0 − sP = (1 p(θ))s (1+r) . 1 − 0 Ontheotherhand,patientinvestorsplace s (1+r)/q buyorders,ofwhichafraction p(θ) 0 1 are executed on average. So patient investors expect to increase their bond holding by 14

p(θ)s (1+r)/q units. Itfollowsthatexpectedconsumptioninthefinalperiodequals 0 1 s (1+r) cP = (1 p(θ))s (1+r)2 + b +p(θ) 0 Rb . (12) 2 − 0 0 q 1 " # That is, the payout from units of storage that were not traded away in the secondary marketplustheexpectedpayoutfromcorporatedebtholdings. 2.4.3 OptimalPortfolioAllocation In the initial period investors solve a portfolio allocation problem, choosing between storageandbondstomaximizetheirexpectedlifetimeutility U = δ(cI +βcI)+(1 δ)cP , 1 2 − 2 subjecttotheperiod0budgetconstraint(9),andtheexpressionsforexpectedconsumption ofimpatientandpatientinvestors(10)-(12). Wecanrewritetheexpectedlifetimeutilityas U = U s +U b , s 0 b 0 whereU andU denotetheexpectedutilityfrominvestinginstorageandbondsinperiod s b 0,respectively,andaregivenby Rb U = δ(1+r)+(1 δ) (1 p(θ))(1+r)2 +p(θ)(1+r) , (13) s − − q 1 " # and U = δ f(θ)q +β(1 f(θ))Rb +(1 δ)Rb . (14) b 1 − − (cid:16) (cid:17) Notethatbothoftheseexpressionsdependonthecharacteristicsofthefinancialcontract, (l ,ωˉ), through the expected return on holding the bond to maturity Rb; and on the 0 characterisitics of the secondary market, (q ,θ), through the secondary market price q 1 1 andmatchingprobabilities f(θ)and p(θ). Using these definitions, we can express the asset demand correspondence that maximizestheinvestorsportfolioproblemas s = 0, b = e if U < U 0 0 0 s b      s 0 ∈ [0,e 0 ], b 0 = e 0 − s 0 if U s = U b     s = e , b = 0 if U > U  0 0 0 s b         15

That is, when the expected benefit of holding storage in period 0 is dominated by the benefit of holding bonds, then investors will demand only bonds in period 0. On the contrary, if the expected benefit of holding storage is greater than then expected benefit of buying a bond in period 0, then investors will only hold storage in the initial period. Finally, if the expected benefits are equal, investors will be indifferent between investing in storage and bonds initially, and their demands will be an element of the set of feasible portfolio allocations: s ,b [0,e ], such that the total value of assets equal the initial 0 0 0 ∈ endowment (9). Given our assumptions, in equilibrium the portfolio allocation will be interior(i.e., U = U with s ,b > 0),thuswefocusouranalysisonthiscase. s b 0 0 Alltold,inequilibriumitmustbethatthetwoassetsinperiod0yieldthesameexpected discountedutility,sothereturntostorageequalsthereturntolendingtoentrepreneurs, U (l ,ωˉ,q ,θ) = U (l ,ωˉ,q ,θ) . s 0 1 b 0 1 Forfuturereferencewelabelthepreviousequationtheinvestors’break-evencondition. Note that the expected utility from investing in storage, U , is not smaller than the expected s utilityinfinancialautarky: U = δ(1+r)+(1 δ)(1+r)2,sincethereturnofbuyingabond a − inthesecondarymarket Δ 1+r (equation 8). ≥ 2.5 Equilibrium Theequilibriumofthemodelisdefinedasfollows. Definition1(CompetitiveEquilibrium) Wesaythat(l ,ωˉ,θ,q )isacompetitiveequilibrium 0 1 ifandonlyif: 1. Given the outcome in the secondary market (θ,q ), the debt contract is described by (l ,ωˉ) 1 0 thatmaximizesentrepreneurs’returnonequitysubjecttoinvestors’break-evencondition. 2. Marketliquiditycorrespondsto θ = (1 δ)(1+r)s /q /(δb ). 0 1 0 − 3. q isdeterminedviathesurplussharingrule. 1 4. Allagentshaverationalexpectationsaboutq and θ. 1 The equilibrium of the model is described by the entrepreneur’s choice of leverage, l , and risk, ωˉ, to maximize the payoff of the risky investment project. Entrepreneurs’ 0 profitsarehigherwhen l ishigherandwhenthepromisedpayoutislower,thatis,when 0 ωˉ is lower. But entrepreneurs are constrained in their choices of l and ωˉ as they need to 0 offertermsthatmakefinancialcontractsattractivetoinvestors: theinvestors’break-even condition. 16

Entrepreneursareawarethatwhensellinginthesecondarymarket,investorsobtaina price that depends on the contract characteristics. In fact, the price is determined via the sharing rule (equation 8). Substituting the secondary market price in the expressions for theexpectedutilitiesofinvestinginstorageandbonds(equations 13and 14)weget U (θ) = δ(1+r)+(1 δ)(1+r) (1 p(θ))(1+r)+p(θ)Δ , s − − and U (l ,ωˉ,θ) = δ f(θ)Δ 1 +(cid:2)(1 f(θ))β +(1 δ) Rb((cid:3)l ,ωˉ) . b 0 − 0 − − n h i o Itfollowsthattheentrepreneur’sproblemcanbewrittenas max [1 Γ(ωˉ)]Rkl 0 l0,ωˉ − subjectto: U (θ) = U (l ,ωˉ,θ) . (15) s b 0 Let λ be the multiplier on the break-even condition (15), then the entrepreneur’s privatelyoptimalchoiceofleverageisgivenby ∂U (l ,ωˉ,θ) [1 Γ(ωˉ)]Rk = λ b 0 . (16) − − ∂l 0 That is, the marginal increase in profits from higher leverage for entrepreneurs need to be proportional to the marginal reduction in expected utility of financial contracts for investors. Similarly,theprivatelyoptimalchoicefortheriskprofileofcorporatedebtisgivenby ∂U (l ,ωˉ,θ) b 0 Γ (ωˉ)l = λ . (17) 0 0 ∂ωˉ That is, the marginal increase in profits from lower risk for entrepreneurs need to be proportionaltothemarginalincreaseinexpectedutilityoffinancialcontractsforinvestors. Takingaratiooftheequations(16)and(17)gives 1 Γ(ωˉ) ∂U (l ,ωˉ,θ)/∂l b 0 0 − = . (18) Γ (ωˉ)l −∂U (l ,ωˉ,θ)/∂ωˉ 0 0 b 0 This equation, which describes the privately optimal debt contract, taken together withtheinvestors’break-evencondition,givenbyequation(15),andtheexpressionsthat characterizethesecondarymarket(θ,q )provideacompletedescriptionoftheequilibrium 1 ofthemodel. Finally, note that both the price in the secondary market q and secondary market 1 17

liquidity θ can be expressed as a function of the characteristics of the optimal financial contract (l ,ωˉ). In fact, the price is a function of the expected return on holding the bond 0 tomaturity Rb,whichdependson(l ,ωˉ);sowecanwritemarketliquidityas 0 (1 δ)s (1+r) (1 δ)(1+r)Δ(e n (l 1)) 0 0 0 0 θ = − = − − − . (19) δb q δn (l 1)Rb(l ,ωˉ) 0 1 0 0 0 − Thefollowingtheoremestablishestheexistenceanduniquenessofequilibriuminour model. Theorem 1 (Existence and Uniqueness of Competitive Equilibrium) Under the maintained assumptions there exists a unique competitive equilibrium of the model. Furthermore, in theuniqueequilibriumcreditisnotrationed,i.e., Γ (ωˉ) μG (ωˉ) > 0. 0 0 − That is, !(l ,ωˉ,θ,q ) where the optimal contract in the primary market is described by (18), 0 1 ∃ the investors’ break-even condition (15) is satisfied, and both secondary market bond pricing and liquidityareconsistentwiththedecisionsinprimarymarkets,i.e.,theyaregivenbyequations(8) and(19),respectively. As is the case in the canonical CSV model (e.g. Bernanke et al. 1999), the result on existence follows from our assumptions. That is, we have assumed that the return on the entrepreurs’ technology is better than the return on financial assets, including the possibility of secondary market retrading, so entrepreneurs will always be able to offer contractual terms that are attractive to investors. In contrast, while uniqueness is relativelystraightforwardtoestablishintheCSVmodel,ourframeworkiscomplicatedby the endogenity of liquidity. Nevertheless, we are able to establish that even in our setup with feedback effects between outcomes in primary and secondary markets, multiple equilibriadonotobtain. 3 Frictions and the (Ir)relevance of OTC Trade Itisusefultodefineabenchmarkinterestratethatisthereturnonatwo-periodbondthat could be traded in a perfectly liquid secondary market. Naturally, such a contract needs todeliverthesamereturninexpectationasastrategyofinvestingonlyinstoragebothin theinitialandinterimperiods.13 Thisgivesrisetothefollowingdefinition. 13Noarbitrageunderperfectlyliquidmarketsimpliesthattradingatwo-periodbondshouldyieldthe sameexpectedreturnforinvestorstorollingoveroneperiodsafeinvestments,i.e. δ R‘/(1+r)+(1 δ) R‘ = ∙ − ∙ δ (1+r)+(1 δ) (1+r)2. ∙ − ∙ 18

Definition 2 (Liquid Two-period Rate) The liquid two-period rate is defined as the gross interestrateonaperfectlyliquidtwo-periodbond. R‘ (1+r)2 . ≡ The benchmark rate allows us to decompose the total gross return on the financial contract written by the firm into a default and a liquidity premium. In order to do this, expressthetotalcorporatebondpremiumasthegrossreturnofthefirm’scontractrelative to the benchmark rate, Z/R‘. Then, this total premium is decomposed into a component owing to default risk, Z/Rb, and a component owing to liquidity risk, Rb/R‘. With this decomposition, we have the following definitions for the default and liquidity premia, respectively. Definition 3 (Default and Liquidity Premia) The default premium Φd and the liquidity premium Φ‘ onthefirm’sdebtcontractaregivenby Z Rb Φd and Φ‘ . ≡ Rb ≡ R‘ Consequently, the total corporate premium is given by Φt Z/R‘ = Φd Φ‘. These ≡ definitionsprovidesharpcharacterizationsofboththedefaultandliquiditypremia,which areconvenienttohelptraceouttheunderlyingeconomicmechanismsinourmodel. The relationship between the liquidity premium and the investors’ break-even condition, in equilibrium,isdescribedinthenextremark. Remark1(InvestorsBreak-evenConditionandLiquidityPremium) Ifinvestorscorrectly expecttheperiod1bondpricetobedeterminedviathesharingrule,thentheinvestors’break-even condition(15)canbeexpressedas (1+r)2Φ‘ = Rb , (20) withtheliquiditypremiumbeingonlyafunctionofsecondarymarketliquiditygivenby 1 δ+(1 δ) (1 p(θ))(1+r)+p(θ)Δ Φ‘(θ) = − − . (21) 1+r δ f(θ)Δ 1 +(1 f(θ))β +(1 δ) −(cid:2) − − (cid:3) On the other hand, the next prop(cid:2)osition shows that th(cid:3)e default premium, in equilibrium,isanincreasingfunctionof only theriskofthefinancialcontract ωˉ. Proposition 1 (Default Premium and Risk) Under the maintained assumptions, the default premium, Φd, depends only on the risk of the financial contract, ωˉ, and it is strictly increasing in ωˉ. 19

Intuitively,investorsdemandahigherdefaultpremiumforfinancialcontractsthatare more likely to default (i.e., contracts that are more risky, or specify a higher productivity thresholdωˉ forpayingoutthefullpromisedvalue). Themoresubtlepartoftheargument is that leverage does not affect the default premium. This is due to the fact that, for a fixed threshold level, ωˉ, leverage affects both the face value of the contract, Z, and the hold-to-maturity return for investors, Rb, in the same way. So leverage is irrelevant for the default premium, as is the case in the benchmark CSV model, though leverage and riskarejointlydeterminedinequilibrium. Wenowturntoourmainresults. 3.1 A Frictionless Benchmark Our first result, stated in Proposition 2, establishes the conditions under which trade in thesecondarymarketisirrelevant,sothatsecondaryOTCmarketliquidityhasnobearing onthefirm’soptimalcapitalstructure. Proposition2(IrrelevanceofOTCTrade) Underthefollowingconditions,thereisnoliquidity premium,i.e., Φ‘ = 1,implyingthatthemodelcollapsestothebenchmarkcostlystateverification model: 1. Allinvestorsarepatient,sothat δ = 0; 2. Impatientinvestorsdiscountatrate β = 1/(1+r); 3. Impatientinvestorsextracttheirfullvaluefromalltheirsellordersinthesecondarymarket, whichistruefor ψ = 1and e eˉ : f(θ) = 1 ;or 0 0 { ≥ } 4. OTC trade is frictionless, which is true in the limit as ν and patient investors have → ∞ deeppockets,i.e.,n << e (1 δ). 0 0 − The case in which δ = 0 is straightforward. When all investors are patient, there is no needtotradeinsecondarymarkets;investorsonlycareaboutthehold-to-maturityreturn. Liquidityisnotpricedinfinancialcontractsandthemodelcollapsestothestandardcostly state verification (CSV) setup presented in, for example, Townsend (1979) and Bernanke andGertler(1989). The same result obtains for the second case, though for different reasons. When impatient investors discount future consumption at exactly the rate of return that comes from holding a unit of storage, so that β = 1/(1 + r), they will be indifferent between consuminginthefinalorinterimperiod. Thisindifferenceimpliesthattherearenogains from OTC trade. In this case, the liquidity preference shock is immaterial and investors 20

only consider the hold-to-maturity return when buying financial contracts in primary markets. The third case considers the situation in which impatient investors can fully satisfy their liquidity needs in secondary markets. That is, the terms of trade are set such that impatient investors extract the entire surplus, i.e., ψ = 1, and all their sell orders will be executed, given that f(θ) = 1. In this case, as before, liquidity considerations will not factor in the lending decision of investors in primary markets. In turn, f(θ) = 1, requires that there is enough storage at t = 1 that all sell orders can be satisfied, which requires that investors’ endowment is sufficiently large. We derive this threshold for investors endowmentintheproofofProposition 2intheAppendix. The final case considers the situation when trade in the secondary market is not subjecttotradefrictions. Inthiscase,investorsareabletotradealltheirholdings. Sincein equilibriuminvestorsneedtobeindifferentbetweenbondsandstorageex-ante,itmustbe thatΔ = Rb/(1+r). Moreover,giventhatpatientinvestorshavedeeppockets,itmustalso bethattheyareindifferentbetweenholdingstorageandtradingbondsat t = 1,implying that Δ = 1+r. Together, these imply that there is no liquidity premium, Φ‘ = 1, and the modelcollapsestothebenchmarkCSV.14 3.2 OTC Trade in the Secondary Market We now characterize the effects of frictional OTC trade. For the remainder of the paper, we consider only the cases in which trading frictions in the secondary market result in a non-negligible liquidity premium. That is, assume that (i) the probability of being an early consumer is positive, δ > 0; (ii) impatient investors discount future consumption strictlymorethanisimpliedbythestoragerate,i.e., β < 1/(1+r);(iii)impatientinvestors cannotfullysatisfytheirliquidityneedsinsecondarymarkets, ψ < 1or f(θ) < 1;and(iv) OTCtradeisfrictional,andwerestrictattentiontothecasewhere ν < 1.15 Undertheseassumptionswebeginbyestablishingthelinkbetweenimperfectliquidity inthesecondarymarketandtheassociatedliquiditypremium. Lemma 1 (Secondary Market Liquidity and the Liquidity Premium) When secondary market liquidity, θ, is lower, investors require a higher liquidity premium, Φ‘, or equivalently, a 14Inthelimitingcasewhere ν ,shouldwedroptheassumptionofpatientinvestors’deeppockets → ∞ there might not be excess liquidity at t = 1 and the bond price could be lower than Rb/(1+r), as in cashin-the-marketpricingmodels(e.g., ShleiferandVishny, 1992). However, inourmodelwithoutaggregate uncertainty,weshowthepricewillalwaysreflectthevaluation(indifferencecondition)ofeitherpatientor impatientinvestorsat t = 1. Asaresult, withoutthedeeppocketsassumptionfirmleverageinitiallywill berationedbytheavailableresourcesofinvestors. 15γ<1guarenteesthatp(θ), f(θ)<1foranyθ. 21

higher hold-to-maturity return, Rb. Moreover, the elasticity of the liquidity premium, Φ‘, with respecttosecondarymarketliquidity, θ,islowerthan 1inabsoluteterms. Lemma 1 formalizes the intuition that the price of liquidity risk (i.e., the liquidity premium)isinverselyproportionaltotheamountofliquidityinsecondaryOTCmarkets. This relationship forms the basis for the direct link between primary and secondary marketsshownbytheupperarrowinFigure1. Inourmodel,marketliquiditydetermines the likelihood that investors’ orders will be executed in an OTC trade. In particular, as the market becomes less liquid sell orders will be more difficult to execute (i.e., f(θ) decreases),andimpatientinvestorswillhaveahardertimefulfillingtheirliquidityneeds insecondarymarkets. Bythesametoken,asliquiditydeclinesbuyordersaremorelikely tobeexecuted(i.e., p(θ)increases)whichprovidesanincentiveforinvestorstoshifttheir portfoliosoutofstorageandintoilliquidbonds. Bothofthesechannelsleadtoareduction in the demand for illiquid bonds in the primary market and an increase in the price of liquidity. In equilibrium, the firm naturally responds to higher funding costs by altering the contract that it issues. A key contribution of this paper is to show that this, in turn, has knock-oneffectsforliquidityinthesecondarymarket(thelowerarrowinFigure 1). This transmissionmechanismissummarizedbythefollowingremark. Remark 2 (The Optimal Contract and Secondary Market Liquidity) Secondary market liquidity, θ, is decreasing in leverage, l , and the riskiness of the contract offered in the primary 0 market, ωˉ. TakentogetherwithLemma 1thisremarkcompletesthefeedbackloopattheheartof this paper. Intuitively, when investors require additional compensation to bear liquidity risk,thefirmhasanincentivetoalterthecharacteristicsofthecontractitoffersinprimary markets, reducing leverage and risk. By doing this, the firm’s actions indirectly enhance liquidityinthesecondarymarket,attenuatingtheinitialincreaseintheliquiditypremium. Similarly,anexogenousshockintheprimarymarketwillripplethroughsecondarymarket liquidity,affectingtheliquiditypremium,andthus,feedingbackintothedecisionsinthe primarymarket. Now we describe the effect of the parameters that determine demand and supply in theprimarymarketintheequilibriumofthemodel. Webeginbydescribingtheeffecton thedemandforbonds. Proposition 3 (Investors’ Bond Demand) Investors require a higher a higher liquidity premium, Φ‘,andhenceahigherhold-to-maturityreturnonthebond,Rb,when 1. (Liquidityshock)Theprobabilityofbecomingimpatientishigher,i.e., δ ishigher; 22

2. (Impatience)Impatientinvestorsdiscountthefuturemoreheavily,i.e., β islower;and 3. (Endowments)Investorshavelesstoinvestinstorage,i.e.,e islower. 0 The proposition describes how the parameters that describe investors’ preferences (δ andβ)andendowments(e )affectdemandintheprimarymarketwhenthecharacteristics 0 ofthefinancialcontract(leverageandrisk)areheldconstant. Asinvestors’preferencesare moresensitivetoliquidityrisk(δishigherorβislower),theassociatedliquiditypremium drives up the hold-to-maturity return that investors require to hold corporate debt. On theotherhand,wheninvestorsarepoorer(e issmaller)theyreducetheirsavingsthrough 0 storage one-for-one conditional on buying the same number of financial contracts. Less liquidsavingsreducesliquidityinsecondarymarkets,andthusalsodrivesuptherequired hold-to-maturityreturnthroughanincreaseintheliquiditypremium(Lemma 1). The equilibrium implications for the optimal capital structure, considering the feedbackloopwithsecondarymarketliquidity,aresummarizedinthefollowingproposition. Proposition4(EquilibriumComparativeStatics) Inequilibrium,thefirm’soptimalleverage, l ,andriskofthecontractsitoffersintheprimarymarket, ωˉ,bothdecreasewhen 0 1. (Liquidityshock)Theprobabilityofbecomingimpatientishigher,i.e., δ ishigher; 2. (Impatience)Impatientinvestorsdiscountthefuturemoreheavily,i.e., β islower; 3. (Investors’Endowments)Investorshavelesstoinvestinstorage,i.e.,e islower;and 0 4. (Firms’Endowments)Firmshavemoreequity(i.e.,n ishigher). 0 This proposition presents the comparative statics in equilibrium for the parameters that describe preferences and endowments for investors and firms. For the first three cases, Proposition 3 establishes that an increase in δ or a decrease in β or e will push up 0 the firm’s cost of funding through the liquidity premium. According to Proposition 4, entrepreneurs adjust to this increase in the cost of funding along two margins (recall that the debt contract is two-dimensional). They offer fewer contacts in the primary market and the contracts that are offered are less risky relative to an equilibrium in which the firm’sdebtistradedwithalowerliquiditypremium. Areductioninthenumberofbonds issued in the primary market lowers the number of possible sell orders in the secondary market, which attenuates the increase in the liquidity premium. That is, the adjustment of the firms’ optimal capital structure mitigates the effect of trading frictions on the price ofliquidity. The fourth case of Proposition 4 deserves special attention. In the benchmark CSV model,alteringthefirm’sendowmentofequityhasnoimpactonthecharacteristicsofthe 23

optimal contract. The reason is because, given an increase in equity, the firm expands it sizeproportionallysothattheoptimalamountofleverage,l = k /n ,remainsunchanged. 0 0 0 Thisresultdoesnotcarrythroughinourframeworkwithendogenoussecondarymarket liquidity. As in the benchmark model—indeed, for exactly the same reason—there is no directeffectofanincreaseinequityontheoptimalcontract. Butourframeworkisdifferent in that an increase in equity raises the number of debt contracts issued in the primary market. To see this consider the firm’s budget constraint expressed in terms of leverage; b = n (l 1). Inorderforl toremainunchanged,thefirmmustincreaseprimaryissuance 0 0 0 0 − in proportion to the size of the equity injection. But, this alters liquidity because it raises the number of possible sell orders in the secondary market, which investors will price through the liquidity premium. Thus, in our framework equity influences the capital structureofthefirm indirectly byalteringsecondarymarketliquidity. Finally, we note that the link between the liquidity premium and the optimal capital structureofthefirmhasthefollowingcorollary. Corollary 1 (Default Premium Comparative Statics) In equilibrium, the default premium Φd decreaseswhen 1. (Liquidityshock)Theprobabilityofbecomingimpatientishigher,i.e., δ ishigher; 2. (Impatience)Impatientinvestorsdiscountthefuturemoreheavily,i.e., β islower; 3. (Investors’Endowments)Investorshavelesstoinvestinstorage,i.e.,e islower;and 0 4. (Firms’Endowments)Firmshavemoreequity,i.e.,n ishigher. 0 ThiscorollaryisadirectconsequenceofPropositions 1and 4. 3.3 A Numerical Illustration We present a simple numerical illustration using the following parameter values. We set the initial endowment of entrepreneurs at n = 0.2 and the endowment of investors at 0 e = 1. Investors’ preferences are described by a discount factor for impatient investors 0 β = 0.85, while δ will take different values in [0,1] to illustrate the results established above. Entrepreneurs’expectedreturnisgivenby Rk = 1.2,whereasthereturnonstorage is assumed to be r = 0.01. The parameters of the matching function in the OTC market arethescalingconstant ν = 0.2andtheelasticityofthematchingfunctionwithrespectto sell orders is α = 0.5. The surplus that accrues to impatient investors in the sharing rule is ψ = 1. Idiosyncratic productivity shocks ω are distributed according to a log-normal 24

distributionwithmeanequal1andvarianceequalto0.25. Finally,monitoringcostsarea share μ = 0.2offirms’revenue. We begin with the frictionless benchmark, taking δ = 0.16 The equilibrium of the model is described by entrepreneurs’ choice of leverage, l , and risk, ωˉ, subject to the 0 constraint imposed by investors’ break-even condition and the consistency requirements for liquidity, θ, and price, q , in the secondary market. The characteristics of the optimal 1 contract (l ,ωˉ) determine the hold-to-maturity return, Rb, and thus the secondary market 0 price q . (Recall that the return on executed orders in secondary markets is pinned down 1 by ψ, r, and β.) The optimal contract will determine the portfolio allocation of investors and thus secondary market liquidity θ (equation 19). Thus, we use the (l ,ωˉ)-space 0 to describe the optimal contract and the equilibrium of the model. Figure 4 depicts the firm’sisoprofitcurvesingreen.17 Investors’break-evenconditionisshownbytheredline. Firm’sprofitsincreasewithleverageanddecreasewithrisk,soisoprofitcurvesrepresent higher profits moving south-east in the figure. The private equilibrum in the frictionless benchmark economy is given by the tangency between the break-even condition and the isoprofitlineshownbythesolidblackdotinFigure 4. Figure 5 illustrates the case of an increase in the liquidity shock, δ, (i.e., the case 1 of Propositions 3 and 4). As the probability of becoming impatient increases, investors require a higher liquidity premium to compensate for liquidity risk (Proposition 3). In contrast, the firm’s isoprofit lines for a given contract specified by (l ,ωˉ) are invariant to 0 δ. Nevertheless, the firm adjusts the terms of the contract it offers in the primary market owing to the increase in the liquidity premium. In particular, the firm reduces its supply of primary debt, which partially compensates investors for the reduction in secondary market liquidity. The resulting equilibrium has a lower level of leverage and a less risky debtcontract,asshowninFigure 5(Proposition 4). Finally, Figure 6 presents a decomposition of the total corporate premium Φt paid on the primary debt contract in terms of the default premium Φd and the liquidity premium Φ‘. The figure shows that lower levels of leverage and risk due to increased liquidity demand result in lower total corporate bond premia. Naturally, the liquidity premium goesup,butthedefaultpremiumdecreasessincethefirmisofferingalowerωˉ (Corollary 1),andthelattereffectdominatesinthiscase. 16FromProposition2thefrictionlessbenchmarkisobtainedifalternativelywesetβ=1/1.01,orifψ=1 (asinourexample)ande issufficientlyhighso f(θ)=1. 0 17Notethattheshapeoftheisoprofitcurves(increasingandconcave)holdsingeneral,asfollowsfrom thepropertiesoftheΓ(ωˉ)function,anddoesnotdependontheparticularvaluesusedinourexample. 25

4 The Efficient Structure of Corporate Debt We analyze the efficient structure of corporate debt by considering a social planner constrainedbythepresenceofmatchingfrictionsandthestructureoftradeinthesecondary market. Hence,ourconceptofefficiencyisoneofconstrainedefficiency,orsecondbest.18 The planner chooses the optimal contract to maximize the profits of the firm while internalizing the effect of the capital structure on secondary markets through liquidity and bond prices. To formalize the planner’s problem let (l ,ωˉ,θ,q ) be allocations that 0 1 describe the socially efficient outcome and let (lce,ωˉce,θce,qce) be the allocations in the 0 1 competitive equilibrium described in section 3. Then, the planner’s problem can be writtenas max [1 Γ(ωˉ)]Rkl (22) 0 ωˉ,l0,θ,q1 − subjectto: U(l ,ωˉ,θ,q ) U(lce,ωˉce,θce,qce) (23) 0 1 ≥ 0 1 andequations(8)and(19). Condition(23)saysthattheplannercannotchooseequilibriumallocationsthatresultin lower welfare for investors compared to the competitive equilibrium, whereas equations (8) and (19) force the planner to respect the determination of prices and liquidity, respectively, in secondary markets.19 The social planning problem differs from the competitive equilibrium in two respects: (1) the planner need not respect the investor’s break-even condition(15),butmaywanttoinfluenceittosatisfy(23);and(2)theplannerinternalizes how period 0 choices affect liquidity in the secondary market by explicitly considering (19) as a constraint, which, in contrast, is an equilibrium condition in the competitive economy.20 Wesubstituteequations(8)and(19)intheplanner’sproblem,andletλbethemultiplier onconstraint(23),toobtainthatthesociallyoptimalchoiceofleverageisgivenby ∂U ∂U∂θ [1 Γ(ωˉ)]Rk = λ n (U U )+b b + . (24) 0 b s 0 − − − ∂l ∂θ ∂l 0 0 " # 18Intheinterestofspacetheanalysisinsections 4and5restrictsattentiontothemoreinterestingcase whereθ (θ,θ),sotradingprobabilitiesdependonthematchingfunction(5)andarenotpinneddownby ∈ theminimumnumberofbuyorsellorders. 19In an Online Appendix we present a more general problem, where the planner can additionally determine the terms of trade in the secondary market and assigns Pareto weights on the two agents to maximizeasocialwelfarefunction. 20Recallthatinvestors,andthusfirms,explicitlyconsidered(8)inthecompetitiveequilibriumaswell, thusitsexplicitconsiderationdoesnotmodifytheplanner’sproblemrelativetothecompetitiveequilibrium, unlesstheplannercanaffectthetermsofsecondarytrade. 26

Thatis,themarginalincreaseinthefirm’sprofitsfromadditionalleverageneedstobe proportional to the marginal reduction in total expected utility for investors. The latter hasthreecomponents: (i)theportfoliocompositioneffect: asleverageincreasesinvestors need to re-allocate n units from storage to bonds; (ii) the effect on the expected utility 0 of bond holdings U ; and (iii) the effect through secondary market liquidity: as liquidity b increasesitbecomeseasierforimpatientinvestorstoselltheirbonds,butitbecomesmore difficultforpatientinvestorstobuybondsandearnthereturn Δinthesecondarymarket. Similarly,thesociallyoptimalchoicefortheriskprofileofcorporatedebtisgivenby ∂U ∂U ∂θ l Γ (ωˉ)Rk = λ b b + . (25) 0 0 0 ∂ωˉ ∂θ ∂ωˉ " # That is, the marginal increase in the firm’s profits from reducing risk need to be proportional to the marginal reduction in total expected utility for investors, which has two components: the effect on the hold-to-maturity return Rb and the effect through secondarymarketliquidity. Takingaratioofequations(24)and(25)gives 1 − Γ(ωˉ) = n 0 (U b − U s )+b 0 ∂ ∂ U l0 b + ∂ ∂ U θ ∂ ∂ l θ 0 . (26) Γ 0 (ωˉ)l 0 − b 0 ∂ ∂ U ωˉ b + ∂ ∂ U θ ∂ ∂ ω θ ˉ This equation, together with the constraint on investors total expected utility (23), describes the socially optimal debt contract.21 We are ready to establish the generic inefficiencyofthedebtcontractincompetitivemarkets.22 Proposition5(GenericConstrainedInefficiencyoftheDebtContract) Consideraplanner that designs an optimal debt contract, as described in (23), (26), (8) and (19). If the parameters (α,ψ,r) belong to a generic set , the planner will set a level of secondary market liquidity that P is different from the competitive equilibrium. That is, the competitive equilibrium is generically constrainedinefficient. GivenProposition 5,wecanidentifytwodistortedmarginsthatdriveasetofwedges between the private and socially efficient outcomes. Comparing the equilibrium conditions (15) and (18) to the social planner’s counterparts (23) and (26), the first distortion is evidentfromthe ∂U/∂θterminequation(26)thatdoesnotappearinequation(18). This term captures the liquidity externality. It arises because neither the firm nor investors 21Theconstraintwillalwaysbebindingsincetheplannercaresonlyaboutthefirm,butthisneednotbe thecaseiftheplannermaximizesaggregatesocialwelfare. Inthatcasetheplannermaywanttosplitthe aggregategainsaccordingtosomesetofParetoweights. 22See also Geanakoplos and Polemarchakis (1986) for a general characterization of constrained inefficiency. 27

internalize the effect that their decisions in the primary market have on liquidity in the secondary market. This additional term changes the trade-off between risk and leverage fortheplannerrelativetothefirm. To understand the role of the term ∂U/∂θ, which measures the externality of market liquidity on investors ex ante welfare, consider the following reinterpretation of the conditionsthatdeterminetheoptimalcontract. Letthenegativeofriskmeasurethesafetyof the financial contract. Then, firms profits are increasing in both leverage and safety, and the optimality conditions can be reinterpreted as equating the marginal benefit with the marginalcost,intermsofinvestorscompensation,ofincreasingleverageorsafety. Using thisinterpretation,theplannerfindsthatapositiveexternalityincreasesthecompensation required to increase leverage and reduces the compensation required to increase safety. Consequently,aplannerthatinternalizesthisexternalitywouldreduceleverageandrisk (increasesafety),leadingtohighersecondarymarketliquidityandfirm’sprofits. Theseconddistortionappearsintheoptimalportfoliocompositionofinvestors. Itcan beeasilyseenbycomparingtheweakParetoimprovementconstraint(23)thattheplanner faces to the break-even condition (15) in the competitive equilibrium, i.e., U = U . Since b s U = U +(1 δ)(1+r)(Δ (1+r))p(θ),wecanrewriteequation(23)asn (l 1)(U U ) = s a 0 0 b s − − − − e (1 δ)(1 + r)(Δ (1+r)) p(θce) p(θ) . Written this way, the equation tells us that as 0 − − − longas∂U/∂θ , 0theplannerchoosesadifferentlevelofmarketliquidity,sothatθce , θ, (cid:2) (cid:3) thenU , U . Inthiscase,theexpectedreturnonholdingbondswillnotbeequatedwith b s thereturntostorage,asmustbethecaseinthecompetitiveequilibrium. Thefollowingpropositionsummarizesthelinkagesbetweenthesetwodistortions. Proposition 6 (Constrained Efficient Equilibrium) The constrained efficient allocations can becharacterizedconditionalonthemodelparameters (α,r,ψ) asfollows: • If ψ(1+αr) > α(1+r) then secondary market liquidity generates a positive externality on investors(∂U/∂θ > 0);theplannerimplementsahigherlevelofsecondarymarketliquidity (θ > θce); and the optimal capital structure of the firm is characterized by lower leverage, l < lce,andlessrisk, ωˉ < ωˉce. 0 0 • If ψ(1+αr) < α(1+r) then secondary market liquidity generates a negative externality on investors (∂U/∂θ < 0); the planner implements a lower level of secondary market liquidity (θ < θce); and the optimal capital structure of the firm is characterized by higher leverage, l > lce,andmorerisk, ωˉ > ωˉce. 0 0 • If ψ(1 + αr) = α(1 + r) then there is no externality (∂U/∂θ = 0) and equilibrium is constrainedefficient,i.e., (l ,ωˉ,θ) = (lce,ωˉce,θce). 0 0 28

To understand the intuition behind the proposition, consider first the role that liquidity has on investors’ welfare. On the one hand, an increase in liquidity generates ex ante welfare gains for impatient investors simply because they will find it easier to sell unwanted corporate debt in secondary markets. On the other hand, patient investors suffer welfare loses as it becomes more difficult to earn a higher return by purchasing bonds at adiscountedpriceinthesecondarymarket. Whetherinvestorsareexantebetteroffwith higherliquiditydependsontheparameterizationof(α,r,ψ). Inparticular,thegainstoimpatientinvestorsoutweighthelossestopatientinvestors, makingexanteinvestorsbetteroff,whenψ(1+αr) > α(1+r). Thisoccurswhenthetrade surplus that accrues to impatient investors is sufficiently large relative to the elasticity of thematchingfunction, ψ > (1+r)/(1+αr).23 Or,alternatively,whenthereturntostorageis α sufficientlylowsuchthatr < (ψ α)/(α αψ). Ineithercase,wesaytheliquidityexternality − − ispositivebecauseexanteinvestorsbenefitfromanincreaseinmarketliquidity. How can the planner implement a higher level of liquidity in a way that increases the profitability of firms? Recall from equation (19) that liquidity can be expressed as a function of the characteristics of the firm’s debt contract, θ(l ,ωˉ). Furthermore, we know 0 ∂θ/∂l < 0 and ∂θ/∂ωˉ < 0. So, from the firm’s perspective, the planner can increase 0 secondary market liquidity by directing the firm to take on less leverage, l < lce, and 0 0 writedebtcontractsthatarelessrisky, ωˉ < ωˉce. Profitabilityincreasesbecause,despitethe reduction in scope owing to lower leverage, the firm reduces its overall cost of funding; higherliquiditylowerstheliquiditypremiumandthelessriskynatureofthedebtcontracts lowersthedefaultpremium.24 Increasing liquidity in this way has, by design, implications for the portfolio composition of investors. Specifically, it requires that investors shift out of corporate bonds and intostorage. Atthesametime,increasingsecondarymarketliquiditydepressesthereturn tostorage(giventhat∂p(θ)/∂θ < 0)andincreasesthereturnonbondholdings(giventhat ∂f(θ)/∂θ > 0). So, investors are being asked to shift their portfolios out of higher return corporate bonds and into storage, which offers a lower return. The only way such an outcomecanobtainisiftheexpectedreturnforholdingbondsinthemoreliquidportfolio dominates the expected return from holding storage, so that (U > U ). In other words, b s theonlywaytosupportallocationsthatdeliverhigherliquidityistoviolatethebreakeven condition. 23TheparameterrestrictionisanalogoustotheHosios(1990)rulethatdeterminestheefficientsurplus splitinsearchandmatchingmodelsofthelabormarket. ArseneauandChugh(2012)studytheimplications ofinefficientsurplussharingforoptimallabortaxationinadynamicgeneralequilibriumeconomy. 24Itisinterestingtonotethatbyimplementinghighersecondary marketliquidity,theplannerinessence increases funding liquidity in the primary market by implementing a reduction in the liquidity premium andthusinthetotalbondpremium. 29

Theoppositeintuitionholdswhenψ(1+αr) < α(1+r),sothattheliquidityexternalityis negative and the planner desires less liquidity relative to the competitive equilibrium. In thiscaseitwillimplementhigherbondpremia,butmakethefirmbetteroffbyincreasing firm’sleverage. Finally,intheknife-edgecasewhere ψ(1+αr) = α(1+r)privateliquidity is efficient so that at the margin an increase in liquidity generates gains for impatient investors that are perfectly offset by losses to patient investors and the planner cannot exploittheexternalitytoimproveuponthecompetitiveequilibrium. 4.1 Decentralizing the Efficient Equilibrium A complete set of tax instruments allows us to decentralize the efficient equilibrium. We introduce a marginal tax τs on the return from storage U (τs < 0 corresponds to a s subsidy) and a marginal tax τl on leverage (τl < 0 corresponds to a subsidy). With these tax instruments, the objective of investors becomes U = b U +s U (1 τs)+Ts and the 0 b 0 s − objectiveofthefirmchangesto[1 Γ(ωˉ)]Rkl τlλl +Tl. Thetaxesarefundedinalump- 0 0 − − sum fashion on the same agents, thus Tl = τlλl and Ts = τss U in equilibrium. Also, in 0 0 s order to simplify the exposition note that we have normalized the tax on leverage by the Lagrange multiplier, λ > 0, on the constraint faced by firms in the competitive economy (i.e.,theinvestor’sbreak-evencondition). Proposition 7providesageneralcharacterizationoftheoptimaltaxpolicy. Proposition 7 (Optimal Policy) The planner’s solution can be decentralized by levying distortionary taxes on the portfolio allocation decision of investors and the capital financing decision of firms. Theresultingoptimaltaxesonstorage, τs,andleverage, τl,aregivenby: e U (θce) τs = 0 1 s , (27) b − U (θ) 0 s ! n U ∂Ubτs + ∂Ub ∂θ ∂Ub ∂θ ∂U τl = 0 s ∂ωˉ ∂l0 ∂ωˉ − ∂ωˉ ∂l0 ∂θ (28) b ∂Uhb + ∂θ∂U i 0 ∂ωˉ ∂ωˉ ∂θ wheretheterminsquarebracketsandthedenominatorin(28)arestrictlypositive. CombiningtheinsightsofProposition 7withProposition 6above,itiseasytocharacterize the optimal tax system more specifically. When ψ(1 + αr) > α(1 + r), the liquidity externality is positive so that the planner wants to implement higher liquidity relative to the competitive equilibrium, θ > θce. Accordingly, the optimal tax system needs to be designed in a way that results in investors holding a more liquid portfolio. This can be achieved through a storage subsidy, so that τs < 0. Moreover, the optimal tax system needs to be designed in a way that results in firms issuing fewer debt contracts in the 30

primary market, which can be achieved through a tax on leverage, so that τl > 0. By the samelogic,whenψ(1+αr) < α(1+r),theliquidityexternalityisnegativeand θ < θce. The optimal tax system calls for a tax on storage, τs > 0, and a leverage subsidy, τl = 0. Only intheknife-edgecasewhere ψ(1+αr) = α(1+r)wehavethat τl = τs = 0. 4.2 A Numerical Illustration We continue the numerical example in section 3.3. Recall that in this illustration, ψ = 1. Moreover,becausetheplannerhasthesameobjectiveasthecompetitivefirm,theisoprofits linesarethesameinbothproblems. Figure7showstheplanner’ssolutionandtheprivate equilibrium for two cases: δ = 0 and δ = 0.1. In a frictionless environment (δ = 0), the planner’s solution coincides with the private equilibrium (as we proved in Proposition 2). However, when there is a positive demand for liquidity, δ > 0 and β < (1+r) 1, and − secondary market liquidity is not sufficiently high to guarantee f(θ) = 1, the planner chooses lower leverage and a less risky capital structure, i.e., lower l and ωˉ. The reason 0 is because the planner internalizes the effect of the leverage decision on liquidity in the secondarymarket. Thisinducestheplannertoconsiderasteeperconstraintcomparedto thebreakevenconditionconsideredbycompetitivefirms(wheremarketliquidityistaken as given). As a result, the planner understands how lower leverage and risk improves borrowingtermsonthemargin,whenthetotalsocialcostsaretakenintoaccount. Table 1 shows the change in equilibrium allocations between the competitive and planner’s solutions for δ = 0.1 as ψ moves from 1 to 0. Consistent with the analysis above,theplanner’sallocationscanbereplicatedbyimposingatax(subsidy)onleverage and storage. For ψ = 1, the liquidity externality is positive, implying that liquidity is suboptimally low in the competitive equilibrium. The planner would like to implement a tax on leverage to generate more liquidity in the secondary market. However, as the share of the gains from trade that accrues to impatient investors declines, the size of the liquidityexternalityshrinks. Hence,theplannerislessaggressiveinchoosingtheoptimal combination of leverage tax and storage subsidy, i.e., both τl and τs shrink in absolute value. When the parameterization of ψ satisfies ψ(1 + αr) = α(1 + r), the externality zeros out and the optimal tax system implies τl = τs = 0. For values of ψ below that point,theliquidityexternalitybecomesnegative,sothatliquidityisover-providedinthe competitive equilibrium. Accordingly, the sign of the optimal tax system flips so that leverageissubsidized, τl < 0,andstorageistaxed, τs > 0. 31

5 Quantitative Easing as part of the Optimal Policy Mix Many central banks following the Great Recession of 2007-09 have turned to unconventional monetary policies, such as quantitative easing (QE), to provide further monetary accomodationaftertheyreducedstandardpolicyratestoitsminimumfeasiblelevels. Ultimately,thegoalofQEisforthecentralbanktoinfluencetherealeconomythroughdirect intervention in the markets for certain assets. Our model provides a stylized framework toanalyzetheeffectofthesepolicies. 5.1 Quantitative Easing Policy We model QE through direct purchases by the central bank of long-term illiquid assets (the financial contracts issued by firms and which are retraded by investors in OTC markets, much like Treasuries and Mortgage Backed Securities). These purchases are financed by the issuance of short-term liquid liabilities, referred to as reserves, that offer a return that is at least as large as that offered by the storage technology. This seems a reasonable approximation for the policies implemented by the Federal Reserve during theGreatRecession,wherelendingfacilitiesandassetpurchaseswerefinancedprimarily withredeemableliabilitiesintheformofreserves(seeCarpenteretal. 2013). In period t = 0, and before markets open, the central bank announces the quantity of ˉ bonds, b , it will purchase in period 0 and will hold to maturity. These bond purchases 0 are financed through the issuance of sˉ units of reserves that pay interest rˉ r. Thus, the 0 ≥ centralbankbudgetconstraintinperiod0issimply b ˉ = sˉ . (29) 0 0 Weassumethecentralbankfinancesitselfinperiod1withreservesonly. Thisassumptionpreventsthecentralbankfrominjectingadditionalresourcesintotheeconomyinthe interim period. In order to keep its bond holdings, the central bank needs to roll over its outstanding reserves and pay interest on them in period 1. The central bank will have to borrow an amount equal to (1 + rˉ)sˉ .25 Finally, in period 2 the central bank receives the 0 debtpayoutfromthefinancialcontractandexpends(1+rˉ)2sˉ ininterestandprincipalon 0 outstanding reserves. It is assumed that the central bank allocates reserves evenly across investorswhodemandreservesinagiventimeperiod. The central bank faces three constraints that, taken together, serve to limit the size 25In practice, the long-term assets held by central banks pay interest in the interim period, and in an environmentoflowshort-terminterestratestheseholdingswillgenerateapositivenet-interestincomefor thecentralbank. Butforsimplicityweabstractfromtheseconsiderations. See, forinstance, Carpenteret al. (2013)forestimatesofnet-interestincomefortheFederalReserve. 32

of its QE program. First, we assume that the central bank is at a disadvantage relative to the private sector in monitoring investment projects. It thus needs to pay a higher monitoringcostrelativetoinvestors,denotedby μˉ > μ.26 Thisimpliesthatinexpectation the central bank anticipates receiving Rˉbb ˉ for its asset holdings, with Rˉb the expected 0 hold-to-maturityreturnonfinancialcontractsforthecentralbank,givenby l l Rˉb(l ,ωˉ) = 0 Rk Γ(ωˉ) μˉG(ωˉ) = Rb(l ,ωˉ) 0 Rk(μˉ μ)G(ωˉ) . 0 0 l 1 − − l 1 − 0 0 − − (cid:2) (cid:3) Second, the central bank needs to fully finance its funding cost, i.e., the total interest onreserves,withitsexpectedreturnonassets. Thatis, Rˉb (1+rˉ)2 . (30) ≥ Finally, we assume that investors cannot be made worse off by QE, as we describe in section 5.3. 5.2 Investors’ Problem and Liquidity with QE Inperiod0investorsallocatetheirwealthacrossthreeassets: thestoragetechnology,debt contracts,andreserves. Sothebudgetconstraintat t = 0isgivenby s +sˉ +b = e , 0 0 0 0 with s ,sˉ ,b 0. Following the approach of Section 2, we consider the optimal behavior 0 0 0 ≥ of impatient and patient investors in t = 1 when they arrive with a generic portfolio of storage,reserves,andbonds(s ,sˉ ,b ). 0 0 0 Impatient Investors. By Assumption 3 impatient investors want to consume all their wealthatt = 1. Theycanconsumethepayoutsoftheirliquidassets: (1+r)s +(1+rˉ)sˉ ;in 0 0 addition,theycanconsumetheproceedsfromtheirsellordersintheOTCmarket: q units 1 of consumption for each order executed. Thus, the expected consumption of impatient investorsinperiods1and2,respectively,isgivenby cI = (1+r)s +(1+rˉ)sˉ + f(θ)q b , (31) 1 0 0 1 0 and cI = (1 f(θ))Rbb . (32) 2 − 0 26Consequently, any positive effects of QE would not accrue from enhanced monitoring, as in the delegated monitoring models of Diamond (1984) and Krasa and Villamil (1992), but from its effect on liquiditypremia. 33

Patient Investors. Patient investors only value consumption in the final period and, as a result,arewillingtoplacebuyordersintheOTCmarketbecausethereturnfromdoingso, Δ, is strictly greater than the return on storage, 1 +r. Moreover, it is also the case that the returnonreserves,1+rˉ,isatleastaslargeasthatonstorage,sopatientinvestorsarewilling to allocate liquid wealth to reserves. Accordingly, liquidity provision in the secondary market will depend on the return on OTC trade, Δ, relative to the return on reserves, 1+rˉ. Specifically, if 1+rˉ < Δ patient investors will pledge all their liquid wealth to place buy orders in the OTC market. On the other hand, if 1 +rˉ > Δ patient investors will use theirliquidwealthtobuyhigheryieldingreservesfirstandthenallocatetheremainderof their liquid wealth to placing buy orders in the OTC market. For expositional purposes, we assume throughout the remainder of the paper that 1 +rˉ < Δ (although for the main resultsofthissection—statedbelowinPropositions 8and9—wetraceouttheproofsover theentireparameterspaceofthemodel,whereappropriate). When the anticipated return to OTC trade exceeds the return on reserves, patient investors use (1 + r)s + (1 + rˉ)sˉ units of liquid wealth to place buy orders. A fraction 0 0 p(θ)arematchedallowingpatientinvestorstoexchangeliquidwealthforcorporatedebt, while the 1 p(θ) unmatched portion needs to be reinvested in liquid assets in period − t = 1. Because the central bank needs to finance itself in the interim period, it removes a totalof(1+rˉ)sˉ reservesfromamass1 δofpatientinvestors. Individualreserveholdings 0 − intheinterimperiodforpatientinvestors, sˉP,totals(1+rˉ)sˉ /(1 δ). Allremainingliquid 1 0 − fundsareplacedintotheloweryieldingstoragetechnology,soexpectedstorageholdings attheendof t = 1,sP,equal 1 (1+rˉ)sˉ sP = (1 p(θ))[(1+r)s +(1+rˉ)sˉ ] 0 , 1 − 0 0 − 1 δ − which is strictly positive from Assumption 4. It follows that expected consumption of patientinvestorsequals (1+rˉ)2sˉ (1+r)s +(1+rˉ)sˉ cP = sP(1+r)+ 0 + b +p(θ) 0 0 Rb . (33) 2 1 1 δ 0 q 1 − ( ) Using the optimal behavior of investors in period 1, summarized in equations (31)- (33), we can rewrite the expected lifetime utility as the portfolio weighted average of the utilitiesofthethreeassetsavailableintheinitialperiod: U = U s +U sˉ +U b . s 0 sˉ 0 b 0 As before, the expected utility of investing in storage and bonds, U and U , are given by s b 34

equations(13)and(14),respectively. Ontheotherhand,theexpectedutilityofreservesis givenby rˉ r U = δ(1+rˉ)+(1 δ)(1+rˉ) (1 p(θ))(1+r)+ − +p(θ)Δ . (34) sˉ − − 1 δ (cid:20) − (cid:21) Reserves yield 1 + rˉ for impatient investors. For patient investors, there is additional compensation that comes from the expected return from buy orders in the secondary market, plus the spread between reserves and storage, rˉ r, for the additional reserves − boughtinperiod1.27 WearenowreadytoestablishthelinkbetweenQEandsecondarymarketliquidity. Proposition8(TheRealEffectsofQuantitativeEasing) Quantitativeeasing,i.e.,thesizeof ˉ the bond buying program, b , increases secondary market liquidity θ and, hence, has implications 0 forthefirm’soptimalcapitalstructureandinvestment. The intuition behind this result is straightforward, each bond bought by the central bankwillbeheldtomaturity,reducingthenumberofsellordersinthesecondarymarket. At the same time, thess bonds need to be financed with reserves, which patient investors can use to submit additional buy orders in the secondary market. So, a bond buying programaffectssecondarymarketliquiditydirectlythroughthepurchaseofbondsaswell asindirectlythroughtheliquiditycreatedbyissuingcentralbankreserves. Moreover,Remark2establishesanequilibriumlinkbetweenliquidityandtheoptimalcapitalstructure ofthefirm,whichdeterminesinvestmentbythefirm. 5.3 QE and Optimal Policy To understand the role of QE in the optimal policy mix, we consider a planner who wants to maximize firm profits, but is restricted by the central bank budget constraint, equation(29),andthefinancingconstraint,equation(30). Inaddition,aswiththeplanner in Section 4, we assume the QE program cannot make investors worse off. To write this later constraint, let U(l ,ωˉ,θ,b ˉ ,rˉ) be the expected lifetime utility of investors when the 0 0 equilibrium is described by (l ,ωˉ,θ), with the secondary market price given by (8), and 0 the QE program described by (b ˉ ,rˉ). Similarly, let U(lce,ωˉce,θce) be the expected lifetime 0 0 utility of investors in the competitive equilibrium, when the secondary market price is given by (8). We refer to this planner that have access to QE policies as the central bank. Then,thecentralbank’sproblemcanbewrittenas 27If,1+rˉ>Δ,patientinvestorswillusetheirliquidwealthfirsttobuyreserves,andthenwillusetheir remaining liquid wealth to place buy orders in the OTC market. Proceeding as above we can derive for patientinvestorssP,cP,andU . 1 2 sˉ 35

max [1 Γ(ωˉ)]Rkl (35) 0 l0,ωˉ,θ,bˉ 0,rˉ − subjectto: U(l ,ωˉ,θ,b ˉ ,rˉ) U(lce,ωˉce,θce) (36) 0 0 ≥ 0 andequations(19),(29)and(30). The following proposition characterizes the role of QE as part of the optimal policy mix. Proposition9(QuantitativeEasingasPartoftheOptimalPolicyMix) Theoptimaldesign ofQEconditionalonthemodelparameters (α,r,ψ) isdescribedasfollows: • If ψ(1 + αr) > α(1 + r) , then QE improves upon the constrained efficient allocation; the ˉ optimalQEprogramconsistsofapositivebondbuyingprogram, b > 0,andpayinginterest 0 onreservesthatarestrictlygreaterthanthereturnonstorage, rˉ > r. • Ifψ(1+αr) α(1+r),thenQEdoesnotimproveupontheconstrainedefficientallocation; ≤ theoptimalQEprogramisjust b ˉ = 0and rˉ = r. 0 As long as the liquidity externality is positive (liquidity is suboptimally low in the privateequilibrium),aQEprogramcanleadtoaParetoimprovementovertheconstrained efficientallocationsstudiedinSection 4. Thereasonthisispossibleisbecausethecentral bank can finance its purchases of long-term illiquid corporate debt by issuing liquid liabilities to investors subject to liquidity risk, much like a deposit contract offered by banks. The central bank has an advantage over a typical bank, however, in that it is not subject to runs by investors. In this sense, a central bank that is not subject to liquidity risk effectively enhances the intermediation technology of the economy. This technological improvement can only be realized when there are social gains from raising liquidity. Whentheliquidityexternalityisnegative(liquidityissuboptimallyhighinthe competitive equilibrium), QE is ineffective because the central bank cannot take a short positionintheprimarycorporatedebtmarket. It is useful to point out that the proposition suggests QE is effective when the interest rate on storage is sufficiently low, r < ψ/(1+α αψ).28 Although it is beyond the scope − of this model, these conditions indicate that QE may be an effective policy response in a protractedlowinterestrateenvironment. TheotherissueworthmentioningisthatwhenQEiseffective,theabsenceofconstraints that limit the size of the program could lead to an extreme outcome in which the central bank disintermediates the bond market. That is, if there is nothing holding back the size 28Alternatively,ψ>α(1+r)/(1+αr)orα<ψ/(1+r rψ). − 36

of the program, as long as QE is effective, the optimal policy is for the central bank to buy all the bonds offered by the firm and offer the corresponding amount of reserves to investors, paying rˉ = r. Doing so allows the central bank to replicate the frictionless benchmark of section 3.1. However, as mentioned above, the size of the QE program is limitedinourmodelby: (1)thehighermonitoringcostthatthecentralbankpaysrelative to investors; (2) the fact that the expected return on assets cannot be lower than the total costofreserves;and(3)investorscannotbemadeworseoff. 5.4 A Numerical Illustration Table 2 extends our numerical example to study QE. The table shows the changes in allocationsrelativetothecompetitiveequilibriumforthreedifferenteconomies. Thefirst columnshowsthedecentralizationofthesociallyefficientoutcomes(throughtheleverage tax, τl, and storage subsidy, τs) without QE, the second column shows the effects of QE by itself, and, finally, the third column shows QE in conjunction with optimal tax policy. All cases assume the parameterization α = 0.5, ψ = 0.9, and r = 0.01. We choose this parameterization because it puts the model in a region of the parameter space where QE iseffective,asperproposition9. Inaddition,weassumethatμˉ = 0.3,whichis50%higher thanthebaselinevalueof μ = 0.2. The first column (which, for reference, corresponds to a point half way between the resultsshowninthefirstandsecondcolumnsoftable 1)showsthatinabsenceofQE,the efficientallocationisdecentralizedwithaleveragetax, τl = 0.21,andamodestsubsidyfor storage,τs = 0.04. Byraisingliquidityinthesecondarymarket,andhencedepressingthe − liquiditypremium,theresultingreductioninfundingcostsraisesprofitsby0.14%relative to the competitive equilibrium, leaving the utility of investors unchanged. The second columnpresentsresultswhereweshutdownthetaxsystem,butallowtheplanneraccess toaQEprogram. Evenwhenweshutdownthetaxsystem,sothatτl = τs = 0,theplanner can use QE to achieve an even greater increase in firm profitability without harming investors. The central bank is able to improve the intermediation technology in the economybydirectlyinterveningintheprimarydebtmarket,financingitsbondpurchases through the issuance of reserves (upon which the central bank must pay investors a premiumabovethereturnonstorage). WithQEtheplannercanachieveasimilaroutcome in terms of liquidity, without tax instruments. Finally, the last column of the table shows that QE, by itself, is not a panacea. A planner can do even better by implementing QE in conjunction with tax policy. The way to interpret this last result is that although QE improves the intermediation technology in the economy, it does nothing to remove the underlyingdistortions. 37

Figure 8 shows how the gains to the firm vary with ψ for different levels of the efficiency of the central bank monitoring technology. The thick lines show the case for μˉ = 0.3assumingQEinconjunctionwiththeoptimaltaxsystem(thethicksolidline)and, alternatively, assuming QE alone with no supporting tax system (the thick dashed line). Thethinsolidanddashedlinescorrespondtothesameinformationwhenthemonitoring cost is higher, so that μˉ = 0.2. Finally, the thin dotted line shows the gains to the firm from optimal tax policy alone in absence of QE. There are four things to take from the figure. First,QEisalwaysmoreeffectivewhencombinedwiththeoptimaltaxpolicy(the solid lines are always above the dashed line for the same monitoring cost assumption). Second, the effectiveness of QE is limited by the parameterization of ψ (the dashed lines are downward sloping, so that as the gains from trade that accrue to impatient investors declines, QE becomes less effective). Third, the effectiveness of QE depends importantly on the quality of the central bank’s monitoring technology (the thick lines are below the thin ones, so the worse the technology, the less effective is QE). Finally, there are parts of the parameter space in which QE is ineffective to the point at which a planner would strictlypreferoptimaltaxationtoQE(theregionsinwhichthethickandthindashedlines liebelowthethindottedline). 6 Conclusion We present a model to study the feedback loop between secondary market liquidity and firm’sfinancingdecisionsinprimarycapitalmarkets. Weshowthatimperfectsecondary marketliquidityaccruingfromsearchfrictionsresultsinpositiveliquiditypremia,lower levels of leverage—or equivalently lower debt issuance,—and less credit risk in primary markets. Lower issuance in primary markets enhances liquidity in secondary markets, butthiseffectisnotenoughtooffsettheriseofliquiditypremia. Furthermore, this feedback loop creates externalities operating via secondary market liquidity, as private agents do not internalize how their borrowing and liquidity provision decisions affect secondary market liquidity. This externality changes the trade-off betweenriskandleverageand genericallymakesthecompetitiveequilibriumconstrained inefficient. We show how efficiency can be restored by correcting two distorted margins: one on firms and one on investors. We consider distortionary taxes to correct these distorted margins, but other instruments such as leverage or portfolio restrictions could also be considered (see also Perotti and Suarez, 2011, who propose Pigouvian taxation to addressexternalitiesfromtheunder-provisionofliquidity). Finally,weshowhowunconventional policies like quantitative easing are expected to affect both secondary market liquidity and debt issuance in primary capital markets. By substituting illiquid assets 38

for liquid short-term securities, these policies increase the intermediation capacity of the economy,andundersomecircumstancesmayleadtoanimprovementontheproductive capacity of the economy. Our analysis suggests that these type of policies ought to be implementedinconjunctionwithpoliciestolimitcorporateborrowing. Inourmodelliquidityholdingsbyinvestorscanbeeithertoolowortoohighrelative to the efficient level, with borrowing by firms being too high or too low, respectively. Thisinefficiencyarisesasbothtypeofagentsfailtointernalizehowtheyaffectsecondary market liquidity. The result is similar to other results in the literature of over-borrowing andliquidityunder-supply. However,ourresulthasdifferentpolicyprescriptionsastwo policy tools tools are needed to restore efficiency. This contrasts with previous results, which have just focused on one of these inefficiencies (Fostel and Geanakoplos, 2008; Farhi, Golosov and Tsyvinski, 2009); or where borrowers are also liquidity providers and one policy instrument is enough to restore efficiency (Holmström and Tirole, 1998; Caballero and Krishnamurthy, 2001; Lorenzoni, 2008; Jeanne and Korinek, 2010; Bianchi, 2011). Our result also highlights the posibility of liquidity over-provision as emphasized byHartandZingales(2015)consideringadifferentmechanism. Ourmodelsuggestasetoftestablepredictionsfortherelationshipbetweentheavailabilityofshort-termliquidassetsandliquiditypremia. Inourmodelthereisonlyoneset of investors who participate in OTC markets, but in practice there are many, potentially segmented OTC markets. In this context, the intuition of our model will predict that liquiditypremiaforagivenasset,shouldbeinverselyrelatedtotheliquidityoftheportfolio oftheparticipantsintheOTCmarketforthatasset. Alongtheselines,ourmodelpredicts thatquantitativeeasingfinancedwithbankreservesshouldhaveaneffectontheliquidity premia of all the securities traded in OTC markets where banks are relevant participants, notonlyaffectingtheliquiditypremiaofilliquidassetspurchasedbycentralbanks. Finally, this paper leaves open questions that we are taking on future work. First, we would like to explore the quantitative relevance of the mechanisms described herein. For that we have deliberately stayed very close to the quantitative model of Bernanke et al. (1999), and we are planning to explore the quantitative prescriptions of our model. Second, in practice many different assets are traded in OTC markets, a dimension that we have abstracted from in our analysis but seems important in practice. Future work shouldexploretherelationshipbetweenmarketsegmentationinOTCtradeandsecondary market liquidity (Vayanos and Wang, 2007; Vayanos and Weill, 2008). Two important considerations that we abstracted from will have to be accounted for in this work: what arethestrategicincentivesinsuchanenvironment?,and,howisliquidityallocatedacross thesemarkets? 39

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Appendix A Proofs Proof of Theorem 1: We need to show that there is a unique equilibrium, and that in this equilibrium credit is not rationed. For that, first, we rule out credit rationing equilibria (Part 1). Then, we establish existence of a non-rationed equilibria (Parts 2-3). Finally, we establish the uniqueness(Part4). Part1. Ruleoutcreditrationingequilibrium. Firstofall,notethatfromAssumption 2,ωˉdF(ωˉ)/(1 F(ωˉ)),isincreasingso − ωˉdF(ωˉ) 1 = μ 1 F(ωˉ) − hasonlyoneroot,whichisstrictlypositiveandisdenotedby ωˉˉ > 0. Notethatfromthedefinitionof Γ(ω)andG(ω)itfollowsthatforωˉ > 0 Γ(ωˉ) > 0 , 1 Γ(ωˉ) = P(ω ωˉ)E[ω ωˉ ω ωˉ] > 0 − ≥ − | ≥ 1 > Γ (ωˉ) = 1 F(ωˉ) > 0 , Γ (ωˉ) = dF(ωˉ) < 0 0 00 − − 0 < G(ωˉ) < 1 , μG(ωˉ) < G(ωˉ) < Γ(ωˉ) d(dF(ωˉ)) (A.1) G (ωˉ) = ωˉdF(ωˉ) > 0 , G (ωˉ) = dF(ωˉ)+ωˉ 0 00 dωˉ limΓ(ωˉ) = 0 , lim Γ(ωˉ) = ωˉP(ω ωˉ)+P(ω < ωˉ)E[ωω < ωˉ] = 1 ωˉ 0 ωˉ ≥ | → →∞ limG(ωˉ) = 0 and lim G(ωˉ) = 1 ωˉ 0 ωˉ → →∞ Then > 0 ifωˉ < ωˉˉ ωˉdF(ωˉ) Γ 0 (ωˉ) − μG 0 (ωˉ) = (1 − F(ωˉ)) 1 − μ 1 F(ωˉ)  = 0 ifωˉ = ωˉˉ . − !  < 0 ifωˉ > ωˉˉ Ontheotherhand,theinvestorsbreak-evencondition(15)  definesarelationshipbetweenrisk ωˉ andleveragel thatwecancharacterizeforgivenmarketliquidity, θ,asfollows. Letωˉibec(l )be 0 0 the correspondence that gives the values of risk compatible with the break-even condition for a levelofleverage,thenthesevaluesofriskareimplicitlydefinedby U (θ) = u (θ)Rb l ,ωˉibec(l ) , s b 0 0 h i whereu (θ) δ f(θ)Δ 1+(1 f(θ))β +(1 δ). b − ≡ − − Sinceinvestorsandthefirmtakesecondarymarketliquidity, θ,asgiven,applyingtheImplicit h i 44

FunctionTheoremforanyωˉ , ωˉˉ wehavethat dωˉibec = ∂ ∂ U l0 b = ∂ ∂ R l0 b . (A.2) dl 0 −∂Ub −∂Rb ∂ωˉ ∂ωˉ Infact,fromequation(4)wehavethat ∂Rb Rb ∂Rb Rb[Γ 0 (ωˉ) μG 0 (ωˉ)] = < 0 and = − , (A.3) ∂l −l (l 1) ∂ωˉ Γ(ωˉ) μG(ωˉ) 0 0 0 − − so we can apply the Implicit Function Theorem for ωˉ , ωˉˉ. It follows that the firm will never chooseacontractwithriskωˉ > ωˉˉ,asfirmprofitsaredecreasinginωˉ,andforωˉ > ωˉˉ additionalrisk willreducethereturntoinvestorsandtheywillnotbewillingtoextendadditionalcredit(higher leverage)atthesehigherrisklevels. An equilibrium with ωˉ = ωˉˉ constitutes a credit rationing equilibrium, since the firm cannot increaseleveragebyincreasingtheriskofthecontract. Wewanttoruleoutthatsuchanequilibrium exists. Note that using a similar argument as above we have that for a fixed θ there exists a function libec(ωˉ) that gives the single value of leverage consistent with the break-even condition. 0 Letl ˉˉ = libec(ωˉˉ) > 0,thentherearethreepotentialtypesofcreditrationingequilibria: (i)l < l ˉˉ ;(ii) 0 0 0 0 l = l ˉˉ ;and(iii)l > l ˉˉ . 0 0 0 0 Suppose in equilibrium the firm chooses (l ,ωˉˉ) with l < l ˉˉ . Since l ˉˉ ,ωˉˉ satisfies the IBEC, 0 0 0 0 it must be that u (θ)Rb(l ,ωˉˉ) > U (θ) from equation (A.3). But then the firm can do better by b 0 s (cid:16) (cid:17) lowering(increasing)theriskofthecontract(leverage),whilestillofferingenoughcompensation toinvestorstoholdbonds, sothiscannotbeanequilibrium. Ontheotherhand, ifequilibriumis described by (l ,ωˉˉ) with l > l ˉˉ , then u (θ)Rb(l ,ωˉˉ) < U (θ). Then, investors at t = 0 will allocate 0 0 0 b 0 s alltheirwealthtostorage,whichisacontradictionwith l ˉˉ > 0. 0 Finally,considerthecasewheretheequilibriumisgivenby(l ˉˉ ,ωˉˉ). Thiscontractissuboptimal 0 forthefirmas Γ (ωˉˉ) = (1 F(ωˉˉ)) > 0 , 0 − which is incompatible with the optimality conditions (16) and (17). Intuitively, the firm can give up an infinitesimal amount of leverage for an infinite reduction of risk, so it will never choose thesecontracttermsinequilibrium. Part2. Rewritetheequilibriumconditionsasasingle-valuedequation (ωˉ). H Note that from equation (18), which characterizes the optimal contract in a non-rationing equilibrium,wecanrearrangetoget Γ(ωˉ) μG(ωˉ) Γ (ωˉ) 0 l (ωˉ) = 1+ − . (A.4) 0 1 Γ(ωˉ) Γ (ωˉ) μG (ωˉ) 0 0 − − Inaddition,notethatfromequation(19)wehave (1 δ)(1+r)Δ(e n (l (ωˉ) 1)) 0 0 0 θ(ωˉ) = θ(l (ωˉ),ωˉ) = − − − . (A.5) 0 δRb(l (ωˉ),ωˉ)n (l (ωˉ) 1) 0 0 0 − Finally,usingequations(A.4)and(A.5)wecanexpressthebreak-evenconditionasthezeroofthe function (ωˉ),definedby H (ωˉ) = U (θ(ωˉ)) u (θ(ωˉ))Rb(l (ωˉ),ωˉ) , (A.6) s b 0 H − 45

Part3. Existenceofanon-creditrationingequilibrium. Want to show that (ωˉ) as a zero in (0,ωˉˉ). But since (ωˉ) is continuous, it suffices to show H H that (0) < 0and (ωˉˉ) > 0. H H Consider first the case ωˉ = 0. From equation (A.4) we have that l (0) = 1, and differentiating 0 equation(A.4)weget dl 0 Γ 0 (ωˉ) Γ(ωˉ) μG(ωˉ) [Γ 0 (ωˉ)]2 μ[Γ 0 (ωˉ)G 00 (ωˉ) Γ 00 (ωˉ)G 0 (ωˉ)] = + − + − , (A.7) dωˉ 1 Γ(ωˉ) [1 Γ(ωˉ)][Γ (ωˉ) μG (ωˉ)] 1 Γ(ωˉ) Γ (ωˉ) μG (ωˉ) − − 0 − 0 ( − 0 − 0 ) sol (0) = 1. Also,Γ (0) = 1andG (0) = 0,thusfromequation(4),lim Rb(ωˉ)equals 00 0 0 ωˉ 0 → lim l 0 (ωˉ) Rk[Γ(ωˉ) μG(ωˉ)] = limRk l 00 (ωˉ)[Γ(ωˉ) − μG(ωˉ)]+l 0 (ωˉ)[Γ 0 (ωˉ) − μG 0 (ωˉ)] = Rk . ωˉ → 0 l 0 (ωˉ) − 1 − ωˉ → 0 l 00 (ωˉ) Inaddition,fromequation(A.5)wehave (1 δ)(1+r)Δ(e n (l (ωˉ) 1)) 0 0 0 limθ(ωˉ) = lim − − − = . ωˉ 0 ωˉ 0 δRb(ωˉ)n 0 (l 0 (ωˉ) 1) ∞ → → − Thisimplythatp(θ) = 0and f(θ) = 1,andthus (0) = δ(1+r)+(1 δ)(1+r)2 Rk δΔ 1+1 δ − H − − − h i = δ (1+r) RkΔ 1 +(1 δ) (1+r)2 Rk < 0 , − − − − h i h i wheretheinequalityfollowsfromAssumption 1. Considernowthecaseωˉ = ωˉˉ,inthiscasefromequation(A.4)wehavethat Γ(ωˉ) μG(ωˉ) Γ (ωˉ) 0 lim l (ωˉ) = lim 1+ − = . 0 ωˉ ωˉˉ ωˉ ωˉˉ 1 Γ(ωˉ) Γ 0 (ωˉ) μG 0 (ωˉ) ∞ → → − − Inaddition,fromequation(4) 1 lim Rb(ωˉ) = lim Rk[Γ(ωˉ) μG(ωˉ)] = Rk[Γ(ωˉˉ) μG(ωˉˉ)] . ωˉ ωˉˉ ωˉ ωˉˉ 1 1/l 0 (ωˉ) − − → → − Furthermore,ifleveragedivergestheninvestorsareallocatingalltheirwealthtobondsandnone tostorage,sos (ωˉˉ) = 0,thenitfollowsfromequation(A.5)that 0 (1 δ)(1+r)Δs (ωˉ) 0 lim θ(ωˉ) = lim − = 0 . ωˉ ωˉˉ ωˉ ωˉˉ δRb(ωˉ)n 0 (l 0 (ωˉ) 1) → → − Thisimplythatp(θ) = 1and f(θ) = 0,andthus (ωˉˉ) = δ(1+r)+(1 δ)(1+r)Δ δβ+1 δ Rk[Γ(ωˉˉ) μG(ωˉˉ)] . H − − − − Toshowthat (ωˉˉ) > 0weproceedbycontradiction(cid:2). Suppose(cid:3)that H δ(1+r)+(1 δ)(1+r)Δ < δβ+1 δ Rk[Γ(ωˉˉ) μG(ωˉˉ)] . − − − Then, at ωˉ = ωˉˉ a portfolio with s = 0 and b =(cid:2) e is opti(cid:3)mal for investors, since in this case the 0 0 0 hold-to-maturityreturnofbonds,Rb = (e +n )/e Rk[Γ(ωˉˉ) μG(ωˉˉ)] > Rk[Γ(ωˉˉ) μG(ωˉˉ)]. Moreover, 0 0 0 − − 46

withthisportfolioallocationliquidityequalszero,sothepreviousinequalitiescapturethereturn onstorageandbondinvestments. Sowehavefoundanequilibriumwithcreditrationing, ωˉ = ωˉˉ, whichisacontradiction. Thus,weconcludethat δ(1+r)+(1 δ)(1+r)Δ < δβ+1 δ Rk[Γ(ωˉˉ) μG(ωˉˉ)] , − − − and (ωˉˉ) > 0. (cid:2) (cid:3) H Part4. Uniqueness: Showthat (ωˉ)isstrictlyincreasingin(0,ωˉˉ). H Differentiatingequation(A.6)weobtain d (ωˉ) dU (θ(ωˉ))dθ(ωˉ) s H = dωˉ dθ dωˉ du (θ(ωˉ))dθ(ωˉ) ∂Rb(l (ωˉ),ωˉ)dl (ωˉ) ∂Rb(l (ωˉ),ωˉ) b Rb(l (ωˉ),ωˉ) u (θ(ωˉ)) 0 0 + 0 , 0 b − dθ dωˉ − ∂l dωˉ ∂ωˉ 0 " # Tosignthisderivativenotethat dU s = (1 δ)(1+r)p (θ)[Δ (1+r)] 0 , 0 dθ − − ≤ du and b = δf (θ) Δ 1 β 0 . 0 − dθ − ≥ h i wheretheinequalitiesfollowfrom p (θ) 0, f (θ) 0,and(1+r) Δ β 1. 0 0 − ≤ ≥ ≤ ≤ Notethatwecanexpress dθ ∂θdl ∂θ 0 = + < 0 . (A.8) dωˉ ∂l dωˉ ∂ωˉ 0 Infact,fromthedefinitionofθ,equation(19),andequation(A.3)wehavethat ∂θ θ(e 0 +n 0 ) ∂θ θ[Γ 0 (ωˉ) μG 0 (ωˉ)] = < 0 and = − < 0 , (A.9) ∂l −l (e n (l 1)) ∂ωˉ − Γ(ωˉ) μG(ωˉ) 0 0 0 0 0 − − − wherethefirstinequalityfollowsfromAssumption4,whereasthesecondinequalityfollowsfrom ωˉ < ωˉˉ. Moreover,fromAssumption2forωˉ < ωˉˉ,dl /dωˉ > 0. Infact,evaluatingequation(A.7)and 0 using that Γ (ωˉ) μG (ωˉ) > 0 for ωˉ < ωˉˉ; Γ (ωˉ)G (ωˉ) Γ (ωˉ)G (ωˉ) = d(ωˉh(ωˉ)) (1 F(ωˉ))2 > 0 from 0 − 0 0 00 − 00 0 dωˉ − Assumption2;andΓ (ωˉ) > 0andΓ(ωˉ) < 1from(A.1). Therefore,dθ/dωˉ < 0. 0 Itisjustlefttoshowthat ∂Rb dl ∂Rb 0 + < 0 , (A.10) ∂l dωˉ ∂ωˉ 0 whichisthecaseiff 1 dl 0 Γ 0 (ωˉ) μG 0 (ωˉ) Γ 0 (ωˉ) 1 Γ(ωˉ)dl 0 > (l 1) − = − l > 0 . 0 0 l dωˉ − Γ(ωˉ) μG(ωˉ) 1 Γ(ωˉ) ⇔ Γ (ωˉ) dωˉ − 0 0 − − 47

Substitutingintheexpressionsfor l (ωˉ)and dl0 fromequations(A.4)and(A.7)weget 0 dωˉ Γ(ωˉ) μG(ωˉ) [Γ (ωˉ)]2 μ[Γ (ωˉ)G (ωˉ) Γ (ωˉ)G (ωˉ)] 0 0 00 00 0 1+ − + − 1 Γ (ωˉ)[Γ (ωˉ) μG (ωˉ)] 1 Γ(ωˉ) Γ (ωˉ) μG (ωˉ) − 0 0 − 0 ( − 0 − 0 ) Γ(ωˉ) μG(ωˉ) Γ (ωˉ) [Γ(ωˉ) μG(ωˉ)]μ[Γ (ωˉ)G (ωˉ) Γ (ωˉ)G (ωˉ)] 0 0 00 00 0 − = − − > 0 − 1 Γ(ωˉ) Γ (ωˉ) μG (ωˉ) Γ (ωˉ)[Γ (ωˉ) μG (ωˉ)]2 − 0 − 0 0 0 − 0 Proof of Proposition 1: From the optimal default decision (2) we have that Z = l /(l 1)Rkωˉ. 0 0 − Ontheotherhand, fromequation(4)wehave Rb = l /(l 1)Rk[Γ(ωˉ) μG(ωˉ)]. Then, thedefault 0 0 − − premiumisgivenby ωˉ Φd(ωˉ) = (A.11) Γ(ωˉ) μG(ωˉ) − Takingthederivative: dΦd(ωˉ) 1 ωˉ[Γ 0 (ωˉ) μG 0 (ωˉ)] = − dωˉ Γ(ωˉ) μG(ωˉ) − [Γ(ωˉ) μG(ωˉ)]2 − − whichislargerthanzeroiff Γ(ωˉ) μG(ωˉ) > ωˉ[Γ (ωˉ) μG (ωˉ)] 0 0 − − UsingthedefinitionofΓ(ωˉ) = ωˉ(1 F(ωˉ))+G(ωˉ)andthatΓ (ωˉ) = 1 F(ωˉ),thepreviousinequality 0 − − isequivalentto (1 μ)G(ωˉ) > ωˉμG (ωˉ) 0 − − But the previous inequality follows from 1 μ > 0, G(ωˉ) 0 and G (ωˉ) = ωˉdF(ωˉ) > 0, for any 0 − ≥ ωˉ > 0. Proof of Proposition 2: When there is no need to compensate investors for liquidity risk, the expected return from lending to entrepreneurs is equal to the outside option of holding storage. In other words, the liquidity premium is zero, i.e. Φ‘(θ) = 1, and Rb = (1 + r)2 or k Rk Γ(ωˉ) μG(ωˉ) = (k n )(1+r)2. Thisisequivalenttothebenchmarkcostlystateverification 0 0 0 − − model where investors are only compensated for credit risk. Note that entrepreneurs’ profits do notd (cid:2) ependdirectly (cid:3) onsecondarymarketliquidity. Weproceedbyshowingthat Φ‘(θ) = 1under thethreealternativeconditionstatedinProposition 2. Condition 1: δ = 0. This implies that secondary market liquidity θ , hence p(θ) = 0. Setting → ∞ δ = 0andp(θ) = 0yieldsΦ‘(θ) = 1. Condition2: β = (1+r) 1. Simplesubstitutionyields Φ‘(θ) = 1. − Condition3: ψ = 1and f(θ) = 1. Simplesubstitutionyields Φ‘(θ) = 1. For any given distribution for the idiosyncratic productivity shock ω, the upper threshold ωˉˉ that entrepreneurs can promise to investors under perfect secondary markets is given by Γ (ωˉˉ) μG (ωˉˉ) = 0, this also gives an upper bound on leverage l ˉˉ (see the proof of Theorem 0 0 0 1). Th − is implies a maximum amount of borrowing, b ˉˉ , which is given by the break-even con- 0 dition (b ˉˉ +n )Rk Γ(ωˉˉ) μG(ωˉˉ) = b ˉˉ (1+r)2. In turn this implies a lower bound for investors’ 0 0 0 − endowment, eˉ , such that θ > θ for e > eˉ given that θ = 1 δ)(1+r) e0 +1 l ˉˉ / δq (l ˉˉ 1) 0 (cid:2) (cid:3) 0 0 − n0 − 0 1 0 − (cid:16) (cid:16) (cid:17)(cid:17) (cid:16) (cid:17) 48

isincreasingine andthehighestvalueforq isl ˉˉ /(l ˉˉ 1)Rk Γ(ωˉˉ) μG(ωˉˉ) . 0 1 0 0 − − (cid:2) (cid:3) Condition 4: ν . In this case f(θ),p(θ) 1. In other words all buy and sell orders are → ∞ → executed. NotethatinthiscasethereturninthesecondarymarketΔ˜ = Rb/q willbeendogenously 1 determinedbysupplyanddemandratherthansurplussplitting. Denoteby y [0,1]thefraction ∈ of bonds sold by impatient investors, and x [0,1] the fraction of wealth that patient investors ∈ exchange for bonds in the secondary market. Then, market clearing in the secondary market requiresthat q δyb = (1 δ)(1+r)xs 1 0 0 − Then, consumption for impatient investors at t = 1 and t = 2, respectively, is given by cI = 1 (1 + r)s + q yb and cI = (1 y)b Rb. Similarly, patient investors allocate to storage at t = 1 0 1 0 2 − 0 sP = (1 x)(1+r)s ,andconsumeinperiod2 cP = (1+r)s + b +(1+r)xs /q Rb. Moreover,we 1 − 0 2 1 0 0 1 canwriteinvestor’sbreak-evenconditionequatingtheexpectedutilityfromstorageandbondsat (cid:0) (cid:1) t = 0,with U = δ(1+r)+(1 δ)(1+r) xΔ˜ +(1 x)(1+r) s − − h i and U = δ yΔ˜ 1+(1 y)β +(1 δ) Rb . b − − − n h i o There are four possible cases to consider depending whether patient and impatient investors areindifferentorstrictlyprefertotradeinthesecondarymarket. First, consider that patient investors are indifferent, i.e., x [0,1], and impatient investors ∈ strictly prefer to trade, so y = 1. The former imply that Δ˜ = 1+r. Substituting in the break-even conditionweobtainthatRb = (1+r)2andq = 1+r. Thatis,thereisnoliquiditypremiumandthe 1 modelwouldcollapsetothebenchmarkCSV.Inorderforthistobeanequilibriumthesecondary marketneedstoclear,whichisthecaseif (1+r)δn (l 1) = (1 δ)(1+r)x(e n (l 1)) n (l 1) < e (1 δ) , 0 0 0 0 0 0 0 0 − − − − ⇔ − − whichfollowsfromthepatientinvestors’deep-pocketassumption. Second, consider that both type of investors strictly prefer to trade in the secondary market. In this case x = y = 1. Substituting these in the break-even condition we get that Δ˜ = Rb/(1+r), henceq = 1+r. Inthiscasemarketclearingwillrequirethat 1 e 0 (1+r)δn (l 1) = (1 δ)(1+r)(e n (l 1)) l = (1 δ)+1 , 0 0 0 0 0 0 − − − − ⇔ n − 0 i.e.,firmborrowingisrationedbythe“endowmentofthepatientinvestors”, e (1 δ). Tosupport 0 − thisequilibrium,investors’wealthshouldbe“scarce”andtheequilibriumintheprimarymarket should support, Δ˜ 1+r, in the secondary market, or equivalently Rb (1+r)2. Therefore, the ≥ ≥ firm can choose the lowest level of risk such that at the given leverage, Rb = (1+r)2. But then there is no liquidity premium, i.e., Φ‘ = 1, as in the previous case with the exception that firm’s borrowing is rationed. However, this cannot be an equilibrium as it contradicts the investor’s deep-pocketassumption. Third, consider that impatient investors are indifferent, i.e., y [0,1], and patient investors ∈ strictly prefer to trade, so x = 1. The former imply that Δ˜ = β 1. Substituting in the break-even − conditionwegetthatRb = β 1(1+r),henceq = 1+r. Then,marketclearingrequiresthat − 1 (1+r)δyn (l 1) = (1 δ)(1+r)(e n (l 1)) n (l 1) > e (1 δ) . 0 0 0 0 0 0 0 0 − − − − ⇔ − − 49

In this case, the firm is able to borrow more compared to the second case above, but needs to compensate (impatient) investors for the fact that they get a bigger discount in the secondary market,equaltoβ 1. Inthiscase,thereisaliquiditypremiumΦ‘ = β 1/(1+r) > 1,butimportantly − − it does not depend on secondary market liquidity θ. Note that this case encompasses a situation wherethefirmborrowsallinventors’endowment, b = e ,andthereisnotradeinthesecondary 0 0 market. This is consistent with investors choices in the primary market as the return on bonds equals the return on storage, and the latter dominates the autarky return δ(1 + r) + (1 δ)(1 + − r)2. However, this cannot be an equilibrium as it contradicts the patient investor’s deep-pocket assumption. Comparing cases two and three above, we observe that the firm in case two is borrowing up to the endowment of the patient investors, but faces lower financing cost. This is because the returninsecondarymarketsisthelowest,1+r,andthisispricedintheprimarymarket,through Rb. Altenatively, in case three the firm borrows more than can be financed by the endowment of patient investors, but faces higher financing cost: there is a liquidity premium. In this case, the returnonthesecondarymarketandthusondebtisthehighest. Whetherthefirmwillchoseone or the other will be determined in equilibrium by the trade-off between leverage and financing costthatdependsonfirm’stechnologyandinvestor’spreferencesandendowments. Finally, consider the case that both type of investors are indifferent. Then, it must be that 1+r = Δ˜ = β 1,whichisacontradictionifβ < 1/(1+r). If,ontheotherhand,β = 1/(1+r)thenthe − investor’sbreak-evenconditionimplythat Rb = (1+r)2,oralternativelythatΦ‘ = 1. ProofofLemma1: Wewanttoshowthatthederivativeoftheliquiditypremium wrtliquidityis negative. DenotebyCandDthenumeratoranddenominatorin Φ‘(θ)givenby21. Then, C = δ+(1 δ) (1 p)(1+r)+pΔ > 0 (A.12) − − D = δ fΔ 1+(cid:2)β(1 f) +(1 δ)(cid:3)> 0 − − − h i where the inequalities follow from the fact that probabilities and returns are non-negative. In addition,denoteC andD thederivativesofCandD,respectively,wrtθ. Then θ θ C = ∂C = (1 δ)[Δ (1+r)] dp(θ) 0 θ ∂θ − − dθ ≤ D = ∂D = δ Δ 1 β df(θ) 0 θ ∂θ − − dθ ≥ h i where the inequalities follow from β < 1/(1 + r), equations (6) and (7), and that the matching functionm(A,B)isincreasinginbotharguments. Fromequation(21)wehavethat dΦ‘ 1 C CD θ θ = 0 (A.13) dθ 1+r D − D2 ≤ (cid:20) (cid:21) where the inequality follows from the previously established inequalities: D,C > 0, D 0 and θ ≥ C 0. θ ≤ In the case where θ < θ and ψ > 0, then D > 0, so dΦ‘/dθ < 0. Alternatively, if θ θ, θ ≥ i.e., f(θ) = 1, our assumptions require that ψ < 1. In addition, dp(θ)/dθ < 0. With ψ < 1 and dp(θ)/dθ < 0 then C < 0, so dΦ‘/dθ < 0. Moreover, when θ θ,θ we have that dp(θ)/dθ < 0 θ ∈ and df(θ)/dθ > 0, so dΦ‘/dθ < 0. Therefore, we conclude that dΦ‘/dθ < 0 when OTC trade is (cid:16) (cid:17) relevant,apartfromthecasewereθ < θandψ = 0. Regarding the second part of the Lemma, the elasticity of the liquidity premium, Φ‘, with 50

respecttothesecondarymarketliquidity, θ,iswritten,usingequation(A.13),as: θ dΦ‘ θ C D CD θ θ ζ = = − , (A.14) Φ‘,θ Φ‘ dθ 1+r CD Then ζ < 1requires: Φ‘,θ (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) θ C θ D − CD θ < 1 C CD+θC D θCD > 0 −1+r D2 1+rD ⇔ θ − θ First, lets consider the case where θ θ,θ . In this case, f(θ) = νθ1 α and p(θ) = νθ 1. Thus, − − ∈ df(θ) dp(θ) θ = (1 α)f(θ)andθ = αp(θ)(cid:16). The(cid:17)n, dθ − dθ − C θ = αC+α[δ+(1 δ)(1+r)] 0 (A.15) θ − − ≤ D θ = (1 α)D (1 α)[βδ+(1 δ)] 0 (A.16) θ − − − − ≥ Then, CD+θC D θCD = CD+D αC+α[δ+(1 δ)(1+r)] C (1 α)D (1 α)[βδ+(1 δ)] θ θ − {− − }− − − − − = αD[δ+(1 δ)(1+r)]+C(1 α)[βδ+(1 (cid:8) δ)] > 0 . (cid:9) − − − Second,considerthecasewhereθ < θ. Inthiscase,p(θ) = 1and f(θ) = θ,sodf(θ)/dθ = 1and dp(θ)/dθ = 0. WanttoshowthatD θD > 0. FromaboveD = δ[Δ 1 β]. Then, θ θ − − − D θD = δβ+(1 δ) > 0 . θ − − Finally, consider the case where θ > θ and ψ < 1. In this case, θdf(θ)/dθ = θD = 0 and θ p(θ) = θ 1. Thus,wewanttoshowthat C+θC > 0. Fromabove,θC = θ 1(1 δ)[Δ (1+r)]. − θ θ − − − − Then, C+θC = δ+(1 δ)(1+r) > 0 . θ − Proof of Proposition 3: From the investors’ break-even condition (20), we see that an increase intheliquiditypremium, Φ‘ inducesinvestorstorequireahigherexpectedreturn Rb toinvestin corporate bonds. Hence, the liquidity premium Φ‘ and the hold-to-maturity bond return Rb are proportionaltooneanother. Infact, dRb Rb (1+r)2Φ‘ = Rb = > 0 ⇒ dΦ‘ Φ‘ Forthisproofweconsidertheliquiditypremiumafunctionofbothsecondarymarketliquidity, θ,andmodelparametersδandβ. Thatis,wecanwritetheliquiditypremiumas Φ‘(θ,δ,β). Case1: Effectofδ. Wanttoshowthat dΦ‘ ∂Φ‘ ∂Φ‘ ∂θ = + > 0 dδ ∂δ ∂θ ∂δ From the definition of secondary market liquidity, given in equation (19), and considering the 51

dependenceofsecondarymarketpricingonliquiditypremia,wehavethat ∂θ θ θ dq 1 dRb dΦ‘ θ θ dΦ‘ = = ∂δ −δ(1 δ) − q 1dRbdΦ‘ dδ −δ(1 δ) − Φ‘ dδ − − Usingthisexpressionweget dΦ‘ = ∂ ∂ Φ δ ‘ − δ(1 1 − δ) z Φ‘,θ dδ 1+ζ Φ‘,θ where ζ is the elasticity of the liquidity premium with respect to secondary market liquidity, Φ‘,θ whichisnegativeandstrictlygreaterthan 1(Lemma1). Therefore,1+ζ > 0. − Φ‘,θ It is left to show that ∂Φ‘/∂δ > 0. For that we use the notation introduced in equation (A.12). Inaddition,denoteC andD thederivativesofCandD,respectively,wrtδ. Then δ δ C = ∂C = 1 (1 p)(1+r)+pΔ δ ∂δ − − D = ∂D = fΔ(cid:2) 1+(1 f)β 1(cid:3) δ ∂δ − − − h i Then,fromequation(21)wehavethat ∂Φ‘ 1 C CD δ δ = ∂δ 1+r D − D2 (cid:20) (cid:21) whichisstrictlygreaterthanzeroifandonlyif C D > CD δ δ C [δD +1] > [δC +1 C ]D δ δ δ δ δ − C > [1 C ]D δ δ δ − or 1 (1 p)(1+r)+pΔ > (1 p)(1+r)+pΔ fΔ 1+(1 f)β 1 − − − − − − (cid:2) (cid:3) (cid:2) (cid:3)nh i o 1 > (1 p)(1+r)+pΔ fΔ 1+(1 f)β − ⇔ − − Itiseasytocheckthatafterdistribut (cid:2) ingtermsinthepr (cid:3) ehviousexpressionithefourremainingterms, are a weighted average of terms strictly smaller than 1, with the weights given by the product of probabilities f andpaddingupto1. Infact,β < 1/(1+r)implythat β(1+r) < 1, Δ 1(1+r) < 1, and Δβ < 1 . − Case2: Effectofβ. Wanttoshowthat dΦ‘ ∂Φ‘ ∂Φ‘ ∂θ = + < 0 . dβ ∂β ∂θ ∂β For that we use the notation introduced in equation (A.12). In addition, denote C and D the β β 52

derivativesofCandD,respectively,wrtβ. Then C = ∂C = (1 δ)pΔ2(1 ψ) < 0 , β ∂β − − − and D = ∂D = δ[f(1 ψ)+1 f] = δ(1 ψf) > 0 . β ∂β − − − wheretheinequalitiesfollowfromourassumptionabout δ,ψ,and f(θ). Then, ∂Φ‘ 1 C β CD β = < 0 , ∂β 1+r D − D2 " # asC < 0andD ,C,D > 0. β β Fromthedefinitionofsecondarymarketliquidity,giveninequation(19),andconsideringthe dependenceofthesecondarymarketpriceonliquiditypremia,wehavethat ∂θ θ ∂q 1 ∂q 1 dRb dΦ‘ 1 dΦ‘ = + = θ (1 ψ)Δ+ ∂β −q 1 ∂β ∂RbdΦ‘ dβ − − Φ‘ dβ " # " # Thus, dΦ‘ ∂ ∂ Φ β ‘ − (1 − ψ)Δθ∂ ∂ Φ θ ‘ = dβ 1+ζ Φ‘,θ where ζ is the elasticity of the liquidity premium with respect to secondary market liquidity. Φ‘,θ From Lemma 1 the denominator, 1 +ζ , is strictly positive. But the sign of the numerator is Φ‘,θ ambiguous. The reason is that a higher β on one hand reduces the preference for liquidity by impatient households, i.e., ∂Φ‘/∂β < 0. But on the other hand it increases the secondary market price, q , which pushes market liquidity θ down and liquidity premia up. This second force, 1 represented by the second term in the numerator, depends crucially on the bargaining power of impatient investors: the lower their bargaining power the more important the effect of their valuation,i.e.,β,willbeontheprice. Thenumeratorisnegativeifandonlyif (1 ψ)Δθ[C D CD ] C D+CD > 0 θ θ β β − − − Usingtheexpressionsderivedabovefor C, D, C , D , C ,andD ,wehave θ θ β β (1 ψ)Δθ[C D CD ] C D+CD θ θ β β − − − = (1 ψ)Δαp(1 δ)[Δ (1+r)][δ(fΔ 1+(1 f)β)+(1 δ)] − − − − − − − (1 ψ)Δ(1 α)fδ[Δ 1 β][δ+(1 δ)[(1 p)(1+r)+pΔ]] − − − − − − − +(1 ψ)p(1 δ)Δ2[δ[fΔ 1+(1 f)β]+(1 δ)] − − − − − +δ[f(1 ψ)+1 f][δ+(1 δ)[(1 p)(1+r)+pΔ]] − − − − =(1 ψ)[δ(fΔ 1+(1 f)β)+(1 δ)] (1 α)p(1 δ)Δ2+p(1 δ)(1+r) − − − − − − − n o +(1 ψ)[δ+(1 δ)[(1 p)(1+r)+pΔ]] αfδ+(1 α)fδΔβ − − − − +(1 f)δ[δ+(1 δ)[(1 p)(1+r)+pΔ]](cid:8)> 0 (cid:9) − − − 53

Case3: Effectofe . Wanttoshowthat 0 dΦ‘ < 0 . (A.17) de 0 Notethatinvestors’endowmente affectsliquiditypremiumΦ‘onlythroughitseffectonsecondary 0 marketliquidityθ. Inparticular,ithasaneffectonlythroughs = e b giventhatwehavefixed 0 0 0 − leverageinthisexercise. Thus, dΦ‘ ∂Φ‘ ∂θ ds ∂Φ‘ θ 0 = = < 0 de ∂θ ∂s de ∂θ s 0 0 0 0 wheretheinequalityfollowsfromLemma 1. ProofofProposition4: Case 1: Comparative Statics on δ. Recall that from equation (18) we can rearrange terms to get leverageasafunctionofrisk,l (ωˉ),equation(A.4). Inaddition,fromequation(19)wecanexpress 0 θasafunctionofl (ωˉ),ωˉ,andδ,i.e.,θ(l (ωˉ),ωˉ,δ). Usingtheseexpressions,equilibriumconditions 0 0 boildowntotheinvestors’break-evencondition,whichcanbeexpressedas (1+r)2Φ‘(θ(l (ωˉ),ωˉ,δ),δ) = Rb(l (ωˉ),ωˉ) 0 0 By the Implicit Function Theorem, if the derivative of the previous expression wrt ωˉ is different than0,thenwecandefineωˉ(δ)andcalculateitsderivativefromthepreviousexpression. Wewant toshowthat dωˉ < 0. dδ Fullydifferentiatingwrttoωˉ weobtain ∂Φ‘ ∂θdl dωˉ ∂θdωˉ ∂θ ∂Φ‘ ∂Rb dl dωˉ ∂Rbdωˉ (1+r)2 0 + + + = 0 + ∂θ ∂l dωˉ dδ ∂ωˉ dδ ∂δ ∂δ ∂l dωˉ dδ ∂ωˉ dδ 0 0 ( " # ) Thus, dωˉ H δ = dδ J with ∂Φ‘ ∂θ ∂Φ‘ H = (1+r)2 + δ − ∂θ ∂δ ∂δ ( ) ∂Φ‘ ∂θdl ∂θ ∂Rb dl ∂Rb J = (1+r)2 0 + 0 (A.18) ∂θ ∂l dωˉ ∂ωˉ − ∂l dωˉ − ∂ωˉ 0 0 ( " #) FromProposition3 ∂Φ‘ > 0. Inaddition, ∂δ ∂θ θ = < 0 ∂δ −δ(1 δ) − and∂Φ‘/∂θ < 0fromLemma1. Thus,H < 0. δ Next we want to show that J > 0. For that first recall that from equation (A.8) we have that (∂θ/∂l )(dl /dωˉ) + (∂θ/∂ωˉ) < 0. Second, note that from equation (A.10) we have that 0 0 (∂Rb/∂l )(dl /dωˉ)+∂Rb/∂ωˉ) < 0. 0 0 Therefore, we conclude that J > 0 and dωˉ/dδ < 0. It follows from dl /dωˉ > 0, equation (A.7), 0 thatdl /dδ < 0. 0 54

ProofofCorollary1: Theeffectofanyparameter%onthedefaultpremiumisdescribedby dΦd dΦd∂ωˉ = . d% dωˉ ∂% Since dΦd > 0fromProposition1,theresultonthedefaultpremiumfollowsfromProposition 4. dωˉ Proof of Proposition 5: We want to show that if the competitive equilibrium is constrained efficient,then(α,ψ,r) ,asetofmeasurezero. ∈ ∅ Suppose(lce,ωˉce,θce,qce),thecompetitiveequilibrium,isconstrainedefficient. Since(lce,ωˉce,θce,qce) 0 1 0 1 isacompetitiveequilibriumtheinvestorbreak-evencondition(15)holds, i.e., U = U , andfrom s b equation(18)itmustbethat 1 Γ(ωˉce) ∂U /∂l b 0 − = . lceΓ (ωˉce) −∂U /∂ωˉ 0 0 b Ontheotherhand,since(lce,ωˉce,θce,qce)isconstrainedefficient,fromequation(26)itmustbethat 0 1 [1 − Γ(ωˉce)] = n 0 (U b − U s )+bc 0 e∂ ∂ U l0 b + ∂ ∂ U θ ∂ ∂ l θ 0. lc 0 eΓ 0 (ωˉce) − bce∂Ub + ∂U∂θ 0 ∂ωˉ ∂θ ∂ωˉ UsingthatU = U ,then s b bce∂Ub + ∂U∂θ ∂Ub 0 ∂l0 ∂θ ∂l0 = ∂l0 , bce∂Ub + ∂U∂θ ∂Ub 0 ∂ωˉ ∂θ ∂ωˉ ∂ωˉ whichisthecaseiff ∂U ∂U ∂θ ∂U ∂θ b b = 0 (A.19) ∂θ ∂ωˉ ∂l − ∂l ∂ωˉ 0 0 " # Notethat, ∂U ∂θ ∂U ∂θ b b < 0, (A.20) ∂ωˉ ∂l − ∂l ∂ωˉ 0 0 since ∂U b = U b < 0 and ∂U b = U b [Γ 0 (ωˉ) − μG 0 (ωˉ)] > 0 (A.21) ∂l −l (l 1) ∂ωˉ Γ(ωˉ) μG(ωˉ) 0 0 0 − − where the last inequality follows from Theorem 1; and ∂θ/∂l ,∂θ/∂ωˉ < 0 from equation (A.9). 0 Then,A.19holdsiff∂U/∂θ = 0,whichisthecaseiff ∂U ∂U sce s +bce b = 0 0 ∂θ 0 ∂θ sce(1 δ)(1+r)[Δ (1+r)]p (θce)+bceδ[Δ 1 β]f (θce)Rb = 0 0 − − 0 0 − − 0 α 1 α p(θce) θce sc 0 e(1 − δ)(1+r)[Δ − (1+r)] = f(θce) θ − ce bc 0 eδ[δ − 1 − β]Rb αsce(1 δ)(1+r)[Δ (1+r)] θce = 0 − − (1 α)bceδ[Δ 1 β]Rb − 0 − − Butfromequation(19)θce = (1 δ)(1+r)Δsce/(δbceRb),then − 0 0 55

α[Δ (1+r)] − = Δ Δ[α+(1 α)β] = 1+αr (1 α)[Δ 1 β] ⇔ − − − − ψ α+(1 α)β α(1 β(1+r)) 1+r +(1 ψ)β = − ψ = − ⇔ 1+r − 1+αr ⇔ 1+αr (1 β(1+r)) − ψ(1+αr) = α(1+r) ⇔ Thesetof(α,ψ,r)satisfying(A)is,thus,ofmeasurezero. ProofofProposition6: Part 1. The sign of the externality determines the socially optimal level of secondary market liquidity. Let betheLagrangianoftheplanner’sproblem,whichisgiven L = [1 Γ(ωˉ)]Rkl λ[Uce s U b U ], 0 0 s 0 b L − − − − Fullydifferentiatingandevaluatingatthecompetitiveequilibriumallocation(lce,ωˉce,θce)wehave 0 ∂U d (lce,ωˉce,θce) = λ dθ, L 0 ∂θ where we have substituted the optimality conditions in the competitive equilibrium. Thus, the planner, who internalizes the effect of liquidity on the investor’s utility, would like to increase liquidity in secondary markets when the externality is positive, i.e., ∂U/∂θ > 0, and decrease liquidityiftheexternalityisnegative,i.e., ∂U/∂θ < 0. Part2. Showthatthesignoftheexternalitydependsontherelationshipbetweentheparameters (α,r,ψ). Wanttoshowthat ∂U ψ(1+αr) > α(1+r) > 0. ⇔ ∂θ Infact, α[Δ (1+r)] ψ(1+αr) > α(1+r) Δ > − ⇔ (1 α)[Δ 1 β] − − − αs (1 δ)(1+r)[Δ (1+r)] 0 θ > − − ⇔ (1 α)b δ[Δ 1 β]Rb 0 − − − ∂U ∂U ∂U b s b +s > 0 > 0. 0 0 ⇔ ∂θ ∂θ ⇔ ∂θ Part3. Characterizationoftheefficientcontract. Let ωˉpi(l ) be the function implicitly defined by the Pareto improvement constraint in the 0 planner’sproblem(23). UsingtheImplicitFunctionTheoremandequation(A.20)wehavethat dωˉpi = ∂ ∂ U l0 + ∂ ∂ U θ ∂ ∂ l θ 0 dl 0 −∂U + ∂U∂θ ∂ωˉ ∂θ ∂ωˉ 56

Similarly,usingthenotationintroducedintheproofofTheorem 1,whereωˉibec(l )denotesthe 0 functionimplicitlydefinedbytheinvestors’break-evenconditioninthecompetitiveeconomyfor ωˉ < ωˉˉ. Fromequation(A.2)wehadthat dωˉibec = ∂ ∂ U l0 b dl 0 −∂Ub ∂ωˉ Notethatthecompetitiveequilibriumisafeasiblepointoftheparetoimprovementconstraint, soωˉpi(lce) = ωˉibec(lce). Moreover,notethat 0 0 dωˉpi(lce) dωˉibec(lce) ∂U ∂θ∂Ub ∂θ∂Ub 0 0 = ∂θ ∂ωˉ ∂l0 − ∂l0 ∂ωˉ dl 0 − dl 0 ∂Ubh b ∂Ub + ∂U∂θ i ∂ωˉ 0 ∂ωˉ ∂θ ∂ωˉ h i whereallthederivativesontheRHSareevaluatedat(lce,ωˉce,θce),andweusedthat 0 ∂U(lce,ωˉce,θce) ∂U (lce,ωˉce,θce) ∂U (lce,ωˉce,θce) 0 = n (U (lce,ωˉce,θce) U (θce))+bce b 0 = bce b 0 ∂l 0 b 0 − s 0 ∂l 0 ∂l 0 0 0 Itfollowsfromaboveandequation(A.20)that dωˉpi(lce) dωˉibec(lce) ∂U 0 0 > 0 > 0. dl − dl ⇔ ∂θ 0 0 Then,ifψ(1+αr) > α(1+r),fromPart2,∂U/∂θ > 0,and,thus, dωˉpi(lce) dωˉibec(lce) 0 0 > > 0 dl dl 0 0 wherethelastinequalityfollowsfromequation(A.21). Thatmeanstherearepointsthatarefeasible fortheplannerwhere(l ,ωˉ) << (lce,ωˉce)thatachievehigherprofitsforthefirm,sotheplannerwill 0 0 chooseanallocationwithlowerleverageandrisk. (Notethatbyequation(A.9)thisimplythatthe planerwillsetahighersecondarymarketliquidity: θ > θce.) Similarly,ifψ(1+αr) < α(1+r),fromPart2,∂U/∂θ < 0,so dωˉpi(lce) dωˉibec(lce) 0 0 0 < < dl dl 0 0 That means there are points that are feasible for the planner where (l ,ωˉ) >> (lce,ωˉce) and higher 0 0 firm’sprofits,sotheplannerwillchooseanallocationwithhigherleverageand risk. ProofofProposition7: Part1. Derivingthetaxinstruments. Thefirm’sproblemwithtaxesonstorageandleveragecanbewrittenas [1 Γ(ωˉ)]Rkl τlλl +Tl (A.22) 0 0 − − subjectto U = (1 τs)U (A.23) b s − 57

WewritetheLagrangianforthisproblemas = [1 Γ(ωˉ)]Rkl τlλl +Tl λ[(1 τs)U U ] (A.24) 0 0 s b L − − − − − Then,theoptimalityconditionsare ∂U [1 Γ(ωˉ)]Rk = τlλ λ b (A.25) − − ∂l 0 ∂U Γ (ωˉ)]Rkl = λ b (A.26) 0 0 ∂ωˉ NotethattheFOCfor(ωˉ),equation(A.26),togetherwithequation(A.21)ensuresthatλ > 0,which isnotnecessarilythecasewithequalityconstraints. Andtheoptimalcontractisdescribedby 1 Γ(ωˉ) ∂Ub τl − = ∂l0 − . (A.27) l 0 Γ 0 (ωˉ) − ∂Ub ∂ωˉ Equating the previous expression and equation (26), and using that U U = τsU , we derive b s s − − thetaxonleverage: n U ∂Ubτs+ ∂Ub ∂θ ∂Ub ∂θ ∂U τl = 0 s ∂ωˉ ∂l0 ∂ωˉ − ∂ωˉ ∂l0 ∂θ b ∂Uhb + ∂θ∂U i 0 ∂ωˉ ∂ωˉ ∂θ Theterminsquarebracketsispositivefromequation(A.20). Ontheotherhand,usingequations (A.9)and(A.21)thedenominatorispositiveiff b δ f(θ)Δ 1+(1 f(θ))β +1 δ Rb 0 − − − n h i o s (1 δ)(1+r)[Δ (1+r)]p (θ)θ b δ Δ 1 β f (θ)θRb > 0. 0 0 0 − 0 − − − − − h i Usingthatp (θ)θ = αp(θ)and f (θ)θ = (1 α)f(θ)thepreviousexpressionequals 0 0 − − b δβ+1 δ Rb+αs (1 δ)(1+r)[Δ (1+r)]p(θ)+αb δ Δ 1 β f(θ)Rb > 0, 0 0 0 − − − − − wherethein (cid:8) equalityfo (cid:9) llowsfrom Δ > 1+randΔ 1 > β,sinceβ < 1 h /(1+r). i − On the other hand, the break-even condition of investors with a tax on storage was given by equation(A.23). Combiningitwithconstraint(23)wederivethetaxonstorage: e U (θce) τs = 0 1 s b − U (θ) 0 s ! Part2. Signingthetaxonstorage. Ifψ(1+αr) > α(1+r)thenfromProposition6theplannerwantstoincreasesecondarymarket liquiditysoθ > θce. Thus,thestoragetechnologyissubsidized: τs 0. Infact,thetaxonstorage ≤ isnegativefromequation(27)ifψ < 1andiszeroifψ = 0. On the contrary, if ψ(1+αr) < α(1+r), then the externality is negative, the planner wants to reducesecondarymarketliquidity,and,therefore, τs > 0. Part3. Signingthetaxonleverage. Westartbydescribingthefeasibleallocationsforafirmthatchoosestheoptimalcontractand 58

facesthetheoptimaltaxonstorage,andtheefficientlevelofsecondarymarketliquidity. Thatis, τs isgivenbyequation(27)andθistheonethattheplannerwouldchooseoptimally. Inthiscase wehave (1 τs)U (θ) = 1 e 0 U s (θ) − U s (θce) U (θ) = b 0 U b (lc 0 e,ωˉce,θce)+s 0 U s (θce) − s 0 U s (θ) s s − − b U (θ) b 0 s 0 ! whereweusedthatinthecompetitiveequilibrium U (θce) = U (lce,ωˉce,θce),andbce+sce = e . s b 0 0 0 0 Letsconsiderfirstthecasewhen ψ(1+αr) > α(1+r). Inthiscase∂U/∂θ > 0andθ > θce,then b U (lce,ωˉce,θce)+s U (θce) < b U (lce,ωˉce,θ)+s U (θ) 0 b 0 0 s 0 b 0 0 s Soweconcludethat (1 τs)U (θ) < U (lce,ωˉce,θ) − s b 0 Since ∂U /∂ωˉ > 0, for the leverage of the competitive equilibrium lce a feasible level of risk lies b 0 below the risk in the competitive equilibrium. So the investor’s break-even condition with the optimaltaxandtheefficientlevelofliquiditywillliebelowtheinvestor’sbreak-evenconditionin the competitive problem. Moreover, from equation (A.2) the slope of this constraint at lce, which 0 hasthesameexpressionregardlessofthetax,willbeflatter. Thefirm,then,ifitweretofacethisconstraintwithoutataxonleveragewillchooseahigher leverage,atoddswiththeplanneroptimalprescriptions. Theplannerthenwilldistortthefirm’s decisiontodisincentivizetheuseofleveragebylevyingataxonleverage. Onewaytoseethisis thattheplannerwillintroduceadistortionsuchthatthedistortedisoprofitlinesareflatterinthe (l ,ωˉ)-space. 0 LetΠτ = [1 Γ(ωˉ)]Rkl τlλl +Tl,anddenotebyωˉΠτ (l )thefunctionthatforanyl givesthe 0 0 0 0 − − associated risk level ωˉ along the taxed firm isoprofit line. Then, the Implicit Function Theorem impliesthat dωˉΠτ [1 Γ(ωˉ)]Rk τlλ = − − dl 0 Γ 0 (ωˉ)]Rkl 0 soaflattersloperequiresapositive τl. Usingthesamereasoningweconcludethatif ψ(1+αr) < α(1+r),thenτl < 0. ProofofProposition8: Inthepresenceofquantitativeeasing,firms’borrowingisgivenbyb +b ˉ , 0 0 whereas investors’ lending is given by b . Then from the budget constraint of entrepreneurs we 0 havethatk = n +b +b ˉ ,soinvestors’lendingcanbewrittenintermsofentrepreneursleverage 0 0 0 0 and QE as b = n (l 1 b ˉ /n ). On the other hand, from the investors’ budget constraint, 0 0 0 0 0 − − b + s + sˉ = e , so we can express the amount invested in the storage technology in terms of 0 0 0 0 entrepreneursleverageas s = n (e /n (l 1)). NotethatthesizeoftheQEprogramdoesnot 0 0 0 0 0 − − affecttheamountultimatelyinvestedinstorage,asthebondsthecentralbankpurchasesareoffset with the reserves it takes from investors. Finally, from the central bank’s budget constraint we havethatsˉ = b ˉ . 0 0 Using the previous expressions we can express secondary market liquidity in terms of entrepreneurs leverage and QE, conditional on the interest on reserves relative to the return on the OTCmarket. NotethatthenumberofsellordersisalwaysequaltoA = δb ,asimpatientinvestors 0 willputalltheirbondholdingsforsaleintheOTCmarket. IfΔ > 1+rˉpatientinvestorspledgealltheirliquidassetstoplacebuyordersintheOTCmarket 59

sothenumberofbuyorders B = (1 δ)[(1+r)s +(1+rˉ)sˉ ]/q andmarketliquidityisgivenby 0 0 1 − (1 δ)[(1+r)s +(1+rˉ)sˉ ] (1 δ)Δ (1+r)(e 0 n 0 (l 0 1))+(1+rˉ)b ˉ 0 θ = − 0 0 = − − − (A.28) δb 0 q 1 h δRb n 0 (l 0 1) b ˉ 0 i − − (cid:16) (cid:17) Then, ∂θ (1 δ)Δ(1+rˉ) (1 δ)Δ (1+r)(e 0 n 0 (l 0 1))+(1+rˉ)b ˉ 0 = − + − − − > 0 (A.29) ∂b ˉ 0 δRb n 0 (l 0 − 1) − b ˉ 0 h δRb n 0 (l 0 − 1) − b ˉ 0 2 i (cid:16) (cid:17) (cid:16) (cid:17) Ontheotherhand,when1+rˉ> ΔpatientinvestorsplacebuyordersintheOTCmarketonly using the liquid assets they hold after funding the reserves liquidated by impatient investors, so thenumberofbuyordersB = (1 δ)[(1+r)s δ/(1 δ)(1+rˉ)sˉ ]/q andmarketliquidityisgiven 0 0 1 − − − by θ = (1 − δ)[(1+r)s 0 − 1 − δ δ (1+rˉ)sˉ 0 ] = (1 − δ)Δ (1+r)(e 0 − n 0 (l 0 − 1)) − 1 − δ δ (1+rˉ)b ˉ 0 δb 0 q 1 h δRb n (l 1) b ˉ i 0 0 0 − − Then, (cid:16) (cid:17) ∂θ = Δ(1+rˉ) + (1 − δ)Δ (1+r)(e 0 − n 0 (l 0 − 1)) − 1 − δ δ (1+rˉ)b ˉ 0 ∂b ˉ 0 − Rb n 0 (l 0 − 1) − b ˉ 0 h δRb n 0 (l 0 − 1) − b ˉ 0 2 i (1 δ (cid:16) )Δ[(1+r)e ( (cid:17) 1+rˉ)n (l 1)+(rˉ (cid:16)r)n (l 1)] (cid:17) 0 0 0 0 0 = − − − − − > 0 2 δRb n (l 1) b ˉ 0 0 0 − − (cid:16) (cid:17) wheretheinequalityfollowsfromAssumption 4. Then,∂θ/∂b ˉ > 0. 0 Proof of Proposition 9: We want to show that a planner that has access to QE as an additional policytoolwillonlyuseitwhenthereturnonstorage risstrictlylowerthan(ψ α)/(α αψ),or − − equivalently, when ψ(1+αr) > α(1+r). Let l sp ,ωˉsp,θsp be the allocations chosen by the social 0 planner studied in section 4 and denote by λsp the lagrange multiplier on the constraint of this (cid:16) (cid:17) planner(23). Let betheLagrangianofthecentralbank,whichcanbewrittenas L = [1 Γ(ωˉ)]Rkl λ Uce U(l ,ωˉ,θ(l ,ωˉ,b ˉ ,rˉ),b ˉ ,rˉ) γ (1+rˉ)2 Rˉb ν[r rˉ]+ηb ˉ 0 0 0 0 0 0 L − − − − − − − h i h i whereweareconsideringtheconstraintimposedbythedefinitionofsecondarymarketliquidity (19)writingθ(l ,ωˉ,b ˉ ,rˉ)andwherewehavealreadysubstitutedin sˉ = b ˉ . Anoptimalallocation 0 0 0 0 forthisplannerneedstosatisfythefollowingFOCs: 60

∂ ∂U ∂U∂θ ∂Rˉb (l ) 0 = L = [1 Γ(ωˉ)]Rk+λ + +γ 0 ∂l − ∂l ∂θ ∂l ∂l 0 0 0 0 " # ∂ ∂U ∂U∂θ ∂Rˉb (ωˉ) 0 = L = Γ 0 (ωˉ)Rkl 0 +λ + +γ ∂ωˉ − ∂ωˉ ∂θ ∂ωˉ ∂ωˉ " # ∂ ∂U ∂U ∂θ (b ˉ ) 0 = L = λ + +η 0 ∂b ˉ ∂b ˉ ∂θ ∂b ˉ 0 " 0 0# ∂ ∂U ∂U∂θ (rˉ) 0 = L = λ + 2γ(1+rˉ)+ν ∂rˉ ∂rˉ ∂θ ∂rˉ − " # Note that the size of the bond buying program b ˉ does not affect firm’s profits directly, as the 0 additionalfundsthatthefirmreceivesfromthecentralbank,b ˉ ,areperfectlyoffsetbythereduction 0 in the amount of funds received from investors, b = n (l 1) b ˉ , as long as firm leverage is 0 0 0 0 − − unchanged. ThenextstepistoevaluatetheFOCsattheconstrainedefficientallocation(withoutQE),i.e., l sp ,ωˉsp,θsp,0,r . IfRˉb(l sp ,ωˉsp) (1+r)2thecentralbankcannotimplementQEwithoutviolatingits 0 0 ≤ fundingconstraint(30). SoweconsiderthatweareintheinterestingcasewhereRˉb(l sp ,ωˉsp) > (1+r)2 (cid:16) (cid:17) 0 and the central bank has some scope to offer a higher return on reserves relative to the storage technology. Inthiscasethemultiplierofthisconstraintat l sp ,ωˉsp,θsp,0,r equalszero,i.e.,γ = 0. 0 Moreover,notethatat b ˉ = 0,investors’expectedutility Uhasthesamefunctionalformasinthe 0 (cid:16) (cid:17) caseoftheplannerstudiedinsection4. Similarly,atb ˉ = 0secondarymarketliquidityθ,equation 0 (A.28), is the same function of choice variables as in the case without QE, equation (19). So we concludethattheFOCswrtleveragel andriskωˉ aresatisfiedat l sp ,ωˉsp,θsp,0,r . (Infact,wecan 0 0 useeitherFOCtoobtainthat λ = λsp,fromwheretheotherFOCfollows.) (cid:16) (cid:17) Next,notethat ∂U = sˉ ∂U sˉ = b ˉ ∂U sˉ ∂U l s 0 p ,ωˉsp,θsp,0,r = 0 0 0 ∂rˉ ∂rˉ ∂rˉ ⇒ (cid:16) ∂rˉ (cid:17) Andgiventhat(1+rˉ) = (1+r) < Δfromequation(A.28)wehavethat ∂θ (1 δ)Δb ˉ ∂θ l sp ,ωˉsp,θsp,0,r 0 0 = − = 0 ∂rˉ δRb n (l 1) b ˉ ⇒ (cid:16) ∂rˉ (cid:17) 0 0 0 − − (cid:16) (cid:17) SotheFOCwrtoninterestonreservesrˉistriviallysatisfied,withν = 0. Finally,weneedtoevaluatetheFOC wrtb ˉ at l sp ,ωˉsp,θsp,0,r . Fromthisconditionitfollows 0 0 that ∂U ∂U ∂θ (cid:16) (cid:17) + < 0 η > 0 and b ˉ = 0 ∂b ˉ ∂θ ∂b ˉ ⇒ 0 0 0 To sign ∂U/∂b ˉ +(∂U/∂θ)(∂θ/∂b ˉ ) we proceed to compute these derivatives and evaluate at 0 0 l sp ,ωˉsp,θsp,0,r . One,notethat 0 (cid:16) (cid:17) U(l ,ωˉ,θ,b ˉ ,rˉ) = [e n (l 1)]U +b ˉ U +[n (l 1) b ˉ ]U 0 0 0 0 0 s 0 sˉ 0 0 0 b − − − − 61

Then, ∂U l sp ,ωˉsp,θsp,0,r 0 = U (θsp,r) U l sp ,ωˉsp,θsp = U (θsp) U l sp ,ωˉsp,θsp (cid:16) ∂b ˉ 0 (cid:17) sˉ − b 0 s − b 0 (cid:16) (cid:17) (cid:16) (cid:17) where we used that if interest on reserves are equal to the return on the storage technology then U (θsp,r) = U (θsp),fromequation(34). Ontheotherhand,fromtheconditionsthatdescribethe sˉ s planner’sallocationswehavethat s sp U (θsp)+b sp U l sp ,ωˉsp,θsp = sceU (θce)+bceU lce,ωˉce,θce = e U (θce) 0 s 0 b 0 0 s 0 b 0 0 s (cid:16) (cid:17) e [U (θce) U(cid:16) (θsp)] (cid:17) U l sp ,ωˉsp,θsp U (θsp) = 0 s − s = τsU (θsp) (A.30) ⇒ b 0 − s b sp − s 0 (cid:16) (cid:17) whereweusedthedefintionoftheoptimaltaxonstorage(27)inthelastequality. Then, from the characterization of the optimal tax on leverage in section 4.1 we have that if r > (ψ α)/[α(1 ψ)],orequivalentlyψ(1+αr) < α(1+r),then − − ∂U ∂U ∂θ ∂U ∂θ ∂U τl < 0 n b U (θsp) U l sp ,ωˉsp,θsp + b b < 0 (A.31) ⇔ 0 ∂ωˉ s − b 0 ∂l ∂ωˉ − ∂ωˉ ∂l ∂θ 0 0 " # h (cid:16) (cid:17)i wherewehavesubstituted(A.30)intotheexpressionfortheoptimaltaxonleverage(28). Two,fromProposition8wehadthat∂θ/∂b ˉ > 0andevaluatingequation(A.29)at l sp ,ωˉsp,θsp,0,r 0 0 weget (cid:16) (cid:17) ∂θ l s 0 p ,ωˉsp,θsp,0,r θspe 0 = (A.32) (cid:16) ∂b ˉ 0 (cid:17) e 0 − n 0 l s 0 p − 1 n 0 l s 0 p − 1 Three,notethatifr > (ψ α)/[α(1 ψ)]wehhaveth(cid:16)atfrom(cid:17)iequ(cid:16)ation((cid:17)A.31)that − − ∂U ∂U ∂U ∂θ ∂U ∂U ∂θ ∂U ∂θ ∂U ∂θ ∂U b b b b b n +n < + +n 0 ∂ωˉ ∂b ˉ 0 0 ∂ωˉ ∂b ˉ 0 ∂θ ( − ∂l 0 ∂ωˉ ∂ωˉ ∂l 0 0 ∂ωˉ ∂b ˉ 0) ∂θ Butthetermincurlybracketsevaluatedat l sp ,ωˉsp,θsp,0,r iszero. Infact,usingequations(A.9), 0 (A.21),and(A.32)wehave (cid:16) (cid:17) ∂U ∂θ ∂U ∂θ ∂U ∂θ b b b + +n = − ∂l 0 ∂ωˉ ∂ωˉ ∂l 0 0 ∂ωˉ ∂b ˉ 0 U b sp θsp Γ 0 (ωˉsp) − μG 0 (ωˉsp) − 1 e 0 +n 0 + e 0 = 0 Soweco Γ n ( c (cid:2) ωˉ lu sp d ) e − th μ a G t (ωˉsp) (cid:3)   l s 0 p (cid:16) l s 0 p − 1 (cid:17) − l s 0 p h e 0 − n 0 (cid:16) l s 0 p − 1 (cid:17)i h e 0 − n 0 (cid:16) l s 0 p − 1 (cid:17)i(cid:16) l s 0 p − 1 (cid:17)   ∂U ∂U ∂U ∂θ ∂U b b n +n < 0 0 ∂ωˉ ∂b ˉ 0 ∂ωˉ ∂b ˉ ∂θ 0 0 And since ∂U /∂ωˉ > 0, then ∂U/∂b ˉ + (∂U/∂θ)(∂θ/∂b ˉ ) < 0. Thus, it must be that if r > (ψ b 0 0 α)/[α(1 ψ)]thenη > 0andtheoptimalQEdesignscallsfornotbuyingbonds,i.e., b ˉ = 0. − 0 − Alternatively,whenr < (ψ α)/[α(1 ψ)]wecanfollowthepreviouslineofargumenttoshow − − 62

thatthetaxonleverageispositiveso ∂U ∂U ∂U ∂θ ∂U ∂U ∂θ ∂U ∂θ ∂U ∂θ ∂U b b b b b n +n > + +n = 0 0 ∂ωˉ ∂b ˉ 0 0 ∂ωˉ ∂b ˉ 0 ∂θ ( − ∂l 0 ∂ωˉ ∂ωˉ ∂l 0 0 ∂ωˉ ∂b ˉ 0) ∂θ wherethederivativesareevaluatedat l sp ,ωˉsp,θsp,0,r . Thus, 0 (cid:16) (cid:17) ∂U ∂U ∂θ + > 0 ∂b ˉ ∂θ ∂b ˉ 0 0 since∂U /∂ωˉ > 0. Thatis,thecentralbankwantstobuybonds,so η = 0. Finally,fullydifferentib ating the Lagrangean of the central bank’s problem and evaluating at the constrained efficient L allocationwithoutQE l sp ,ωˉsp,θsp,0,r ,wehavethat 0 (cid:16) (cid:17) ∂U ∂U ∂θ d = λ + db ˉ > 0. L ∂b ˉ ∂θ ∂b ˉ 0 " 0 0# So we conclude that when r < (ψ α)/[α(1 ψ)] a central bank will set positive bond buying program, improving upon the cons − trained effi − cient allocation. When b ˉ it follows from the FOC 0 wrtrˉthatthecentralbankwillpayahigherinterestonreservesrelativetothereturnonthestorage technology. In fact, γ will be strictly positive and the central bank’s funding constraint will be binding. 63

Tables and Figures Table1: PlanningoutcomesandImplementation ψ 1.0 0.8 0.6 0.4 0.2 0.0 %changeinl -8.62% -5.03% -1.63% 1.72% 5.13% 8.63% 0 %changeinωˉ -5.27% -3.06% -0.99% 1.04% 3.08% 5.17% %changeinθ 62.01% 27.75% 7.44% -6.70% -17.42% -26.03% %changeinΠ 0.23% 0.07% 0.01% 0.01% 0.06% 0.16% %changeinU 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% τl 0.27% 0.15% 0.05% -0.05% -0.13% -0.21% τs 0.00% -0.05% -0.03% 0.04% 0.14% 0.27% Note: Percentagescorrespondtodeviationswithrespecttothecompetitiveequilibriumforvariables: leverage(l0),risk(ωˉ),market liquidity(θ),firms’profits(Π),andinvestors’utility(U);andtotheleveloftheoptimaltaxesonleverage(τl)andstorage(τs).Negative valuesfortaxescorrespondstosubsidies.Fordetailsseesection4.2. Table2: OutcomeswithQuantitativeEasing ConstrainedEfficient QuantitativeEasing QuantitativeEasing Allocations withτs =τl =0 withτs,τlChosenOptimally %changeinl -6.78% 1.68% -3.05% 0 %changeinωˉ -4.13% 0.72% -2.35% %changeinθ 42.19% 43.37% 167.72% %changeinΠ 0.14% 0.42% 0.98% %changeinU 0.00% 0.00% 0.00% rˉ 1.16% 1.10% sˉ 0.09 0.18 0 τl 0.21% 0.17% τs -0.04% -0.05% Note: Percentagescorrespondtodeviationswithrespecttothecompetitiveequilibriumforvariables: leverage(l0),risk(ωˉ),market liquidity(θ),firms’profits(Π),andinvestors’utility(U);andtothelevelof:taxonleverage(τl),taxonstorage(τs),andinterestrate onreserves(rˉ).Valuesforreserves(sˉ0)areinlevels.Negativevaluesfortaxescorrespondstosubsidies.Fordetailsseesection5.4. 64

Figure3: CreditMarketInstrumentLiabilities (Nonfinancialcorporatebusiness,millions2013dollars) Source: BalanceSheetofNonfinancialCorporateBusiness(B.103),FinancialAccountsofthe UnitedStates;FederalReserveEconomicData(FRED)St.LouisFed. Notes: The data corresponds to the following series in the Financial Accounts: commercial paper (FL103169100); municipal securities and loans (FL103162000); corporate bonds (FL103163003); loans corresponds to the sum of depository institution loans n.e.c. (FL103168005) and other loans and advances (FL103169005); and total mortgages (FL103165005). Figure4: EquilibriumintheFrictionlessBenchmark Break-even condition Inidifference curves of firm Equilibrium Note:Fordetailsseesection3.3. 65

Figure5: ComparativeStaticson δ. Break-even condition for δ=0 Break-even conditions for δ>0 Inidifference curves of firm Equilibrium Note:δtakevaluesin 0,0.1,...,0.5.Seesection3.3. { } Figure6: BondPremiaDecomposition Impatience (δ) Note:Fordetailsseesection3.3. 66

Figure7: ConstrainedEfficientEquilibrium Break-even condition for δ=0 Break-even condition C.E. for δ>0 Inidifference curves of firm Break-even condition planner for δ>0 C.E. Planning solution Note:Fordetailsseesection4.2. Figure8: EffectofQuantitativeEasing 1.0% 0.8% Optimal Taxes 0.6% Low μ : QE Low μ : QE with Optimal Taxes High μ : QE 0.4% High μ : QE with Optimal Taxes 0.2% 0% 0.98 0.96 0.94 0.92 0.9 0.88 0.86 0.84 0.82 Surplus split (ψ) Note:Fordetailsseesection5.4. 67